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Types of Geometry
Their Relationship to Each Other
The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.
The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy (“betweenness”) but not that of measurement.
An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). Topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.
The Axiomatic Approach to Geometry
See H. G. Forder, The Foundations of Euclidean Geometry (1927); H. S. M. Coxeter, Introduction to Geometry (2d ed. 1969).
a branch of mathematics that studies spatial relationships and forms as well as relationships and forms that resemble spatial ones in structure.
The origin of the term “geometry,” which literally means “earth measurement,” may be explained in the following words, attributed to the ancient Greek scholar Eudemos of Rhodes (fourth century B.C.):“Geometry was discovered by the Egyptians as a result of their measurement of land. This measurement was indispensable to them because of the overflowing of the river Nile, which continually washed out boundaries.” The ancient Greeks already considered geometry to be a mathematical science, while for the science of measuring land the term “geodesy” was introduced. Judging by fragments that have been preserved from ancient Egyptian writings, one can see that geometry developed not only from measurements of land but also from measurements of volumes and surfaces in excavation and construction work.
Initial concepts of geometry arose as a result of the abstraction from bodies of all properties and relationships except mutual location and size. Location is expressed in the contact or adjacency of bodies to one other, in terms that one body is part of another, in the placement “between,” “inside,” and so forth. Size is expressed in the concepts “larger” and “smaller” and in the concept of congruence of bodies.
It is by means of the same kind of abstraction that the notion of a geometric body arises. A geometric body is an abstraction in which only the shape and dimensions are preserved, fully abstracted from all other properties. In this case, geometry, like all of mathematics, totally ignores the indeterminacy and mobility of real shapes and dimensions and regards all the relationships and shapes it investigates as precise and definite. Abstraction leads from the concept of extension of bodies to the concepts of surface, line, and point. This is clearly expressed, for example, in the definitions given by Euclid:“a line is a length without breadth” and “a surface is that which has only length and breadth.” A point without any extension is an abstraction reflecting the possibility of an unlimited reduction of all the dimensions of a body, the imaginary limit of its infinite division. Next there arises the general notion of a geometric figure, by which is understood not only a body, surface, line, or point but any set of them.
Geometry in its original meaning is the science of figures, of the mutual disposition and dimensions of their parts, and of transformations of figures. This definition fully agrees with the definition of geometry as the science of spatial shapes and relationships. Indeed, a figure as it is considered in geometry is a spatial shape; hence in geometry one says, for example, “sphere” and not “a body of spherical shape.” The disposition and dimensions are determined by spatial relationships. Finally, a transformation as it is understood in geometry is also a certain relationship between two figures—the given one and the one into which it is transformed.
In the contemporary, more general meaning, geometry encompasses various mathematical theories whose connection with geometry is determined not only by the similarity (however tenuous at times) of their subject matter to ordinary spatial shapes and relationships but also by the fact that they historically evolved and are evolving on the basis of geometry in its original meaning and in their constructions proceed from an analysis, generalization, and variation of its concepts. Geometry in this general meaning is tightly intertwined with other branches of mathematics and its boundaries are not precise (see the sections Generalization of the subject matter of geometry and Modern geometry).
Development. Four main periods can be distinguished in the development of geometry, the transitions between which signified qualitative changes in geometry.
The first period, that of the formation of geometry as a mathematical science, occurred in ancient Egypt, Babylonia, and Greece and lasted approximately until the fifth century B.C. Primary geometric data appeared in the earliest stages of society’s development. The establishment of the first general principles should be considered the rudiments of a science, and in this case they were the establishment of relationships between geometric quantities. This moment cannot be dated. The earliest writing containing the rudiments of geometry has reached us from ancient Egypt and dates approximately to the 17th century B.C.; undoubtedly it was not the first written account. Geometric data in that period were few and amounted primarily to the calculation of certain areas and volumes. They were set forth in the form of rules, largely empirical in origin it appears, and logical proofs were probably still very primitive. Geometry, according to the testimony of Greek historians, was transmitted from Egypt to Greece in the seventh century B.C. There it evolved into an orderly system over several generations. This process occurred through the accumulation of new geometric knowledge, clarification of relationships between various geometric facts, the working out of methods of proof and, finally, the formation of the concepts of figure, geometric proposition, and proof.
This process finally led to a qualitative leap forward. Geometry was transformed into an independent mathematical science. Systematic presentations of it appeared in which its propositions were successively proved. It is at this time that the second period of the development of geometry begins. There are known references to systematic accounts of geometry, among which was one made in the fifth century B.C. by Hippocrates of Chios. What has been preserved—and later played a decisive role—is Euclid’s Elements, which appeared around 300 B.C. Geometry is presented here basically as it is understood today, if one restricts oneself to elementary geometry: it is the science of the simplest spatial shapes and relationships, developed in logical sequence and proceeding from explicitly formulated basic propositions—axioms and basic spatial concepts. The geometry that has been developed on these very bases (axioms) and even made more precise and enriched both in subject matter and in methods of inquiry is called Euclidean geometry. Already in Greece new results were added to it. New methods of determining areas and volumes (Archimedes, third century B.C.) appeared as well as a theory of conic sections (Apollonius of Perga, third century B.C.). The beginnings of trigonometry (Hipparchus, second century B.C.) and spherical geometry (Menelaos, first century B.C.) sprang up. The decline of classical society led to a comparative stagnation in the development of geometry, but it continued to develop in India, Middle Asia, and the countries of the Arab East.
The renaissance of the sciences and arts in Europe brought with it the further flourishing of geometry. A fundamentally new stride was made in the first half of the 17th century by R. Descartes, who introduced the method of coordinates into geometry. The method of coordinates made it possible to link geometry with algebra, which was developing then, and with analysis, which was just being conceived. The application of the methods of these sciences in geometry gave rise to analytic geometry and later differential geometry. Geometry passed on to a qualitatively new level in comparison with the geometry of the ancients: it already was considering much more general figures and was using essentially new methods.
It is at this time that the third period of the development of geometry begins. Analytic geometry studies figures and transformations that are defined by algebraic equations in rectangular coordinates using the methods of algebra. Differential geometry, which arose in the 18th century as a result of the works of L. Euler, G. Monge, and others, already investigates all sufficiently smooth curves and surfaces as well as families (that is, continuous sets) and transformations of such objects (a more general meaning is often attached today to the concept of “differential geometry”; (see the section Modern geometry). The name differential geometry is associated chiefly with its method, which is based on differential calculus. The birth of projective geometry in the works of G. Desargues and B. Pascal dates to the first half of the 17th century. It arose as a result of problems in representing bodies on a plane. Its first subject matter comprised the properties of plane figures that are preserved upon projection from one plane onto another from any point. These new currents of geometry were given final form and systematic presentation in the 18th and early 19th century by Euler for analytic geometry (1748), by Monge for differential geometry (1795), and by J. Poncelet for projective geometry (1822), although the study of geometric representation (directly connected to problems of mechanical drawing) was developed still earlier (1799) and systematized by Monge as descriptive geometry. In all of these new disciplines, the foundations (axioms, basic concepts) of geometry remained unchanged, but the range of figures under study and their properties, as well as the methods being used, expanded.
The fourth period in the development of geometry was begun by N. I. Lobachevskii’s construction in 1826 of a new, non-Euclidean geometry, which is now called Lobachevskian geometry. The same geometry was constructed by J. Bolyai independently of Lobachevskii in 1832 (the same ideas were developed by K. Gauss, but he did not publish them). The source, essence, and importance of Lobachevskii’s ideas reduce to the following. Euclidean geometry contains a postulate on parallel lines that states that “Through a point not on a given line there passes one and only one line parallel to the given line.” Many geometers attempted to prove this postulate by proceeding from other basic premises of Euclid’s geometry, but they failed. Lobachevskii arrived at the idea that such a proof was impossible. The opposite contention to Euclid’s postulate states: “Through a point not on a given line there pass at least two lines parallel to the given line.” This is Lobachevskii’s postulate. According to Lobachevskii’s idea, adding this statement to other basic statements of geometry leads to logically faultless conclusions. The system of these conclusions was what formed the new, non-Euclidean geometry. Lobachevskii’s contribution consisted in the fact that he not only stated this idea but actually constructed and comprehensively developed a new geometry, which was logically just as complete and rich in results as Euclidean geometry, despite the fact that it went against intuition. Lobachevskii viewed his geometry as a possible theory of spatial relationships; however, it remained hypothetical until its real meaning was clarified (in 1868) and it was thereby fully substantiated (see the section Interpretations of geometry).
The revolution in geometry owing to Lobachevskii is of no less significance than any of the revolutions in natural science, and it was not without reason that Lobachevskii was called “the Copernicus of geometry.” His ideas contained three principles that determined the new development of geometry. The first principle is that it is logically conceivable to have not just Euclidean geometry but other “geometries” as well. The second principle is the principle of the construction of new geometric theories through the modification and generalization of the basic propositions of Euclidean geometry. The third principle is that the truth of a geometric theory, in the sense of its agreement with the real properties of space, can be verified only by physical investigation, and it is not ruled out that such investigations will establish that in this sense Euclidean geometry is imprecise. Present-day physics has confirmed this. However, this does not negate the mathematical validity of Euclidean geometry, since mathematical validity is synonymous with logical consistency. In the case of any geometric theory we must differentiate between its physical truth and its mathematical truth; its physical truth consists in its conformity with reality, which is verifiable by experiment, while its mathematical truth consists in logical consistency. Lobachevskii thus provided a materialist orientation for the philosophy of mathematics. The aforementioned general principles played an important role not only in geometry but also in mathematics in general, in the development of its axiomatic method, and in understanding its relation to reality.
The main feature of the new period in the history of geometry, begun by Lobachevskii, was the development of new geometric theories—new “geometries”—and the corresponding generalization of the subject matter of geometry. The concept of different kinds of “spaces” arose. (The term “space” has two meanings in science: on the one hand it is ordinary, real space and on the other, it is abstract “mathematical space.” In the process, some theories evolved within Euclidean geometry, which only later acquired independent significance. This was how projective, affine, conformal, and other geometries developed, the subject matter of which was the properties of figures that are preserved under the corresponding (projective, affine, conformal, and other) transformations. The concepts of projective, affine, and conformal spaces arose; Euclidean geometry itself began to be viewed in a certain sense as a part of projective geometry. Other theories, like Lobachevskii’s geometry, were constructed from the very beginning on changing and generalizing the concepts of Euclidean geometry. In this way, for example, multidimensional geometry was created. The first works on it (H. Grassmann and A. Cayley, 1844) represented a formal generalization of ordinary analytic geometry from three to n coordinates. In 1872, F. Klein summarized the development of all these new “geometries,” pointing out the common principle of their construction.
A fundamental step was made by G. F. B. Riemann (1854 lecture, published 1867). First, he clearly formulated the generalized concept of space as a continuous set of any homogeneous objects or phenomena (see the section Generalization of the subject matter of geometry). Second, he introduced the concept of a space with any law of measuring distances by infinitesimal steps (similar to the concept of measuring length on a curve using a very small scale). Hence there developed a vast area of geometry, so-called Riemannian geometry, and its generalizations, which found important applications in the theory of relativity, mechanics, and elsewhere.
In the same period topology was born as a theory of those properties of figures that depend only on the adjacency of their parts and that are thereby preserved under all transformations that neither destroy adjacencies nor introduce new ones; to put it briefly, in a topological transformation neither breaks nor fusions can arise. In the 20th century topology has developed into a separate discipline.
Thus geometry has turned into a rapidly developing ramified set of mathematical theories which study various spaces (among them Euclidean, Lobachevskian, projective, and Riemannian spaces) and figures in these spaces.
Concurrently with the development of new geometric theories, the already established fields of Euclidean geometry were being elaborated—elementary, analytic, and differential geometry. At the same time, new currents appeared in Euclidean geometry. The subject matter of geometry also broadened in the sense that the range of figures under study broadened as did the range of their properties and the concept of a figure itself. At the juncture of analysis and geometry a general theory of point sets arose in the 1870’s, which, however, is no longer attached to geometry but constitutes a separate discipline. A figure began to be defined in geometry as a set of points. The development of geometry was closely associated with the profound analysis of the properties of space that underlie Euclidean geometry. In other words, it involved a more exact definition of the foundations of Euclidean geometry itself. This work led in the late 19th century (D. Hilbert and others) to a precise formulation of the axioms of Euclidean geometry, as well as those of other geometries.
Generalization of the subject matter of geometry. It is easiest to clarify the possibility of generalizing and modifying geometric concepts by using an example. Thus, on the surface of a sphere points can be connected by shortest lines, which are the arcs of great circles, angles and areas can be measured, and various figures can be constructed. Their study constitutes the subject matter of spherical geometry, just as plane geometry is geometry on a plane. Geometry on the earth’s surface is close to spherical geometry. The laws of spherical geometry differ from the laws of plane geometry. For instance, the circumference in the former is not proportional to the radius but increases at a diminishing rate and reaches a maximum at the equator. The sum of the angles of a triangle on a sphere is not constant and is always larger than two right angles. Similarly, on any surface one can draw curves, measure their lengths and the angles between them, and determine the areas bounded by them. Surface geometry developed in this way is called its intrinsic geometry (K. Gauss, 1827). On an irregularly curved surface, the ratios of lengths and angles will vary from place to place; consequently the surface will be geometrically inhomogeneous, in contrast to a plane and a sphere. The possibility of obtaining various geometric relationships suggests that the properties of real space can be only approximately described by ordinary geometry. This idea, first expressed by Lobachevskii, found confirmation in the general theory of relativity.
A broader opportunity for generalizing the concepts of geometry is implicit in the following argument. Ordinary real space is regarded in geometry as a continuous set of points, that is, as the set of all possible limiting locations of an infinitely small object. Similarly, a continuous set of possible states of some material system and a continuous set of some uniform phenomena can be treated as a kind of “space.” Here is an example. Experiments show that normal human vision is trichromatic, that is, any sensation of color C is a combination—the sum of three basic sensations: red R, green G and blue B, with certain intensities. Designating these intensities in some units as x, y, z, one can write C = xR + yG + zB. Just as a point can be moved in space up and down, right and left, forward and backward, so the sensation of color C can be continuously changed in three directions by changing its components—red, green, and blue. By analogy, one can say that the set of all colors is a three-dimensional space—“a space of colors.” The continuous change of color can be represented as a curve in this space. Furthermore, if two colors are given, for example, red R and white W, then by combining them in various proportions a continuous sequence of colors can be obtained, which may be called a segment RW. The notion that pink P lies between red and white and that a dark pink lies closer to red does not require explanation. Thus, there arise the concepts of the simplest “spatial” shapes (line, segment) and relationships (between, closer to) in the space of colors. Next, one can introduce a precise definition of distance (for example, in terms of the number of thresholds of distinction that can be drawn between two colors), define surfaces and domains of colors, like ordinary surfaces and geometric objects, and so forth. In this way a theory of the space of colors arises, which by generalizing geometric concepts reflects the real properties of man’s chromatic vision.
Another example. The state of a gas in a cylinder under a piston is determined by pressure and temperature. The set of all possible states of a gas can therefore be represented as a two-dimensional space. The “points” of this “space” are the states of the gas; the “points” are distinguished by two “coordinates”—pressure and temperature—just as points in a plane are distinguished by the values of their coordinates. A continuous change of state is depicted by a curve in this space.
Next, one can imagine any material system—mechanical or physicochemical. The set of all possible states of this system is called its phase space. The “points” of this space are the states themselves. If the state of a system is determined by n quantities, then it is said that the system has n degrees of freedom. These quantities play the role of coordinates of the point-state; in the example involving a gas the role of coordinates was played by pressure and temperature. Accordingly, such a phase space of the system is said to be n-dimensional. A change in state is depicted by a curve in this space; individual domains of states, which are distinguished by certain characteristics, are domains of the phase space, and the boundaries of the domains are surfaces in this space. If the system has only two degrees of freedom, its states can be depicted as points in a plane. Thus, the state of a gas with pressure p and temperature T is depicted by a point with coordinates p and T, while the processes occurring in the gas will be depicted by curves in the plane. This method of graphic depiction is universally known and is constantly used in physics and technology for a visual representation of processes and their laws. But if the number of degrees of freedom is more than three, simple graphic depiction (even in space) becomes impossible. Then, in order to preserve useful geometric analogies, one has recourse to the idea of an abstract phase space. Thus, visual graphic methods develop into this abstract idea. The method of phase spaces is widely used in mechanics, theoretical physics, and physical chemistry. In mechanics the motion of a mechanical system is represented by the motion of a point in its phase space. In physical chemistry it is especially important to examine the form and contiguity of those domains of a phase space of a system of several substances that correspond to qualitatively different states. The surfaces separating these domains are surfaces of transition from one state to another (melting, crystallization, and so forth). In geometry itself, one also considers abstract spaces whose “points” are figures. We thus have “spaces” of circles, spheres, lines, and so forth. In mechanics and the theory of relativity an abstract four-dimensional space is introduced by adding time as a fourth coordinate to the three spatial coordinates. This means that events must be distinguished not only by their position in space but also in time.
Thus, it becomes clear how continuous sets of various objects, phenomena, or states can be brought under a generalized concept of space. In such a space one can draw “curves” depicting continuous sequences of phenomena (or states); draw “surfaces” and determine in an appropriate way “distances” between “points,” thereby giving a quantitative expression to the physical concept of the degree of difference between the corresponding phenomena (or states); and so forth. In this way, by analogy with ordinary geometry, a “geometry” of abstract space arises. The latter may bear little resemblance to ordinary space, being, for example, in-homogeneous in its geometric properties and finite—similar to an irregularly curved, closed surface.
It turns out that the subject matter of geometry in a generalized sense is not only spatial forms and relationships but any forms and relationships that, when abstracted from their content, prove to be similar to ordinary spatial forms and relationships. These space-like forms of reality are called spaces and figures. Space in this sense is a continuous set of similar objects, phenomena, or states, which play the role of points and are connected by relationships similar to ordinary spatial relationships, such as distance between points, congruence of figures, and so forth (a figure, in general, is part of space). Geometry views these forms of reality in abstraction from concrete content. The study of concrete forms and relationships in connection with their qualitatively distinct content forms the subject matter of other sciences, and geometry serves as a method for them. An example can be any application of abstract geometry, even the aforementioned application of n-dimensional space in physical chemistry. Geometry is characterized by an approach to an object that consists in generalizing and transferring ordinary geometric concepts and visual representations to new objects. This is precisely what is done in the example of the space of colors and other examples cited above. This geometric approach is not at all mere convention but corresponds to the very nature of phenomena. However, the same real facts can often be depicted analytically or geometrically, just as one and the same function can be expressed by an equation or by a curve on a graph.
The development of geometry should not be perceived, however, as though it merely records and describes in geometric language space-like forms and relationships that have already been encountered in practice. In reality, geometry defines broad classes of new spaces and figures in them, proceeding from an analysis and generalization of the data of intuitive geometry and already established geometric theories. In an abstract definition, these spaces and figures act as possible forms of reality. Consequently, they are not purely speculative constructions but must ultimately serve as a means of investigating and describing real facts. Lobachevskii, in creating his geometry, regarded it as a possible theory of spatial relationships. And just as his geometry was substantiated in terms of its logical consistency and applicability to natural phenomena, so any abstract geometric theory undergoes the same dual verification. The method of constructing mathematical models of new spaces is of fundamental value for verifying logical consistency. However, the only abstract concepts that permanently take root in science are those that are justified both by the construction of an artificial model and by applications, if not directly in natural science and technology then at least in other mathematical theories through which these concepts in one way or another are linked to reality. The ease with which mathematicians and physicists now operate with various “spaces” has been achieved as a result of the long development of geometry in close association with the development of mathematics as a whole and of other exact sciences. It was precisely as a consequence of this development that the second aspect of geometry, mentioned in the general definition at the beginning of the article, evolved and took on great significance: the incorporation into geometry of the investigation of forms and relationships similar to forms and relationships in ordinary space.
As an example of an abstract geometric theory, one can consider the geometry of n-dimensional Euclidean space. It is constructed by means of a simple generalization of the basic propositions of ordinary geometry, for which there are several possibilities: one can, for instance, generalize the axioms of ordinary geometry, but one can also proceed from assigning coordinates to points. In the second approach, n-dimensional space is defined as a set of some elements called points, each of which is determined by an ordered n-tuple of numbers x1, …xn—its coordinates. Next, the distance between the points X = (x1, x2, …, xn and X’ = (x1’, x2’ …, xn’) is defined by the formula
which is a direct generalization of the well-known distance formula in three-dimensional space. A motion is defined as a transformation of a figure such that the distances between its points do not change. Then the subject of n-dimensional geometry is defined as the study of those properties of figures that do not change under motions. On this basis it is easy to introduce concepts of a straight line, of planes of various dimensions from two to n - 1, of a sphere, and so forth. A theory develops in this way that is rich in content and largely analogous to ordinary Euclidean geometry but also largely different from it. It frequently happens that results obtained for three-dimensional space are easily carried over, after appropriate changes, into a space of any number of dimensions. For example, the theorem that of all objects with the same volume a sphere has the smallest surface area reads exactly the same way for a space of any number of dimensions [one must only have in mind n-dimensional volume, (n - 1)-dimensional area, and an n-dimensional sphere, which are defined completely analogously to the corresponding concepts of ordinary geometry]. Furthermore, in n-dimensional space the volume of a prism is equal to the product of the area of the base and the height, while the volume of a pyramid is equal to the same product divided by n. There are many such examples. On the other hand, in multidimensional spaces qualitatively new facts are also found.
Interpretations of geometry. Any one geometric theory allows for different applications and different treatments (realizations, models, interpretations). Any application of a theory is in fact nothing more than a realization of some of its inferences in an appropriate field of phenomena.
The possibility of different realizations is a common property of all mathematical theories. For example, arithmetic relations are realized in the most diverse sets of objects; the same equation often describes completely different phenomena. Mathematics considers only the form of a phenomenon, abstracted from the content, and in terms of form many qualitatively different phenomena often turn out to be similar. The diversity of the applications of mathematics and of geometry in particular is assured precisely by the abstract nature of mathematics. A system of objects (domain of phenomena) is considered to provide a realization of a theory if the relations in this domain of objects can be described in the language of the theory so that every assertion of the theory expresses a certain fact occurring in the domain that is being examined. In particular, if a theory is constructed on the basis of some system of axioms, then the interpretation of this theory consists in such a juxtaposition of its concepts with certain objects and their relationships that the axioms are fulfilled for these objects.
Euclidean geometry arose as a reflection of the facts of reality. Its usual interpretation, in which taut threads are considered lines, a mechanical displacement is considered a motion, and so forth, preceded geometry as a mathematical theory. The question of other interpretations was not raised and could not be raised until a more abstract understanding of geometry emerged. Lobachevskii created non-Euclidean geometry as a possible geometry, and then the question of its real interpretation arose. This problem was solved in 1868 by E. Beltrami, who noted that Lobachevskii’s geometry coincided with the intrinsic geometry of surfaces of constant negative curvature, that is, the theorems of Lobachevskii’s geometry describe geometric facts on such surfaces (where the role of straight lines is taken by geodesies and the role of motions by the bending of the surface onto itself). Since at the same time such a surface is an object of Euclidean geometry, it turned out that Lobachevskii’s geometry was being interpreted in terms of Euclid’s geometry. In this way the consistency of Lobachevskii’s geometry was proved, since a contradiction in it, by dint of the foregoing interpretation, would entail a contradiction in the geometry of Euclid.
Thus the dual significance of interpreting a geometric theory—physical and mathematical—is clarified. An interpretation using concrete objects yields an experimental proof of the truth of the theory (with corresponding precision, of course). If the objects themselves are of an abstract nature (say, a geometric surface in the framework of Euclidean geometry), then the theory is linked to another mathematical theory, in this case to Euclidean geometry, and through it to the experimental data summed up in it. This kind of interpretation of one mathematical theory by means of another has become a mathematical method for substantiating new theories, a technique for proving their consistency, since a contradiction in a new theory would give rise to a contradiction in the theory by which it is being interpreted. But the theory by means of which the interpretation is being effected needs substantiation itself. Therefore, the foregoing mathematical method does not eliminate the fact that the final criterion of truth for mathematical theories is experience. At present, geometric theories are most often interpreted analytically. For example, points in a Lobachevskii plane can be linked to pairs of numbers x and y, straight lines can be defined by equations, and so forth. This technique substantiates a theory because mathematical analysis itself is ultimately substantiated by the enormous volume of its applications.
Modern geometry. The formal mathematical definition of the concepts of space and a figure that is used in modern mathematics stems from the concept of a set. Space is defined as a set of some elements (“points”) subject to the condition that certain relationships similar to ordinary spatial relationships are established in this set. A set of colors, a set of states of a physical system, or a set of continuous functions defined on a segment [0, 1] form spaces where the points are colors, states, or functions. More precisely, these sets are thought of as spaces only if the appropriate relationships—for example, the distance between points—and the properties and relationships defined through them are incorporated in them. Thus, the distance between functions can be defined as the maximum of the absolute value of their difference: max |f(x) - g(x) |. A figure is defined as an arbitrary set of points in a given space. Sometimes a space is a system of sets of elements. For example, in projective geometry it is customary to view points, lines, and planes as coequal fundamental geometric objects that are linked by a relationship of “join.”
The principal types of relationships, which in various combinations bring about the entire diversity of “spaces” of modern geometry, follow.
(1) The general relationships that exist in any set are those of membership and inclusion: a point is a member of a set, and one set is part of another. If only these relationships are taken into consideration, then no “geometry” is yet defined in the set, and it does not become a space. However, if some special figures (sets of points) are distinguished, then the “geometry” of the space can be defined by the laws that connect the points with these figures. In elementary, affine, and projective geometry, this role is played by the axioms of incidence; here straight lines and planes serve as special sets.
The same principle of distinguishing some special sets makes it possible to define the concept of a topological space—a space in which “neighborhoods” of points are distinguished as special sets (subject to the conditions that a point belongs to its own neighborhood and each point has at least one neighborhood; the imposition of further requirements on a neighborhood defines particular types of topological spaces). If every neighborhood of a given point has common points with a certain set, then such a point is called a closure point of that set. Two sets may be called contiguous if at least one of them contains closure points of the other. A space or figure will be continuous, or, as it is said, connected, if it cannot be broken down into two noncontiguous parts. A transformation is continuous if it does not upset contiguities. Thus, the concept of a topological space is used to give a mathematical description of the concept of continuity. A topological space may also be defined by other special sets (closed, open) or directly by the relationship of closure which associates to a set the set of its closure points.
Topological spaces as such, the sets within them, and their transformations serve as the subject matter of topology. To a large extent, the subject matter of geometry proper comprises the investigation of topological spaces and figures in them that are endowed with additional properties.
(2) The second highly important principle of defining and investigating various spaces is the introduction of coordinates. A manifold is a (connected) topological space in which one can introduce coordinates in the neighborhood of each point by placing the points of the neighborhood in a one-to-one bicontinuous correspondence with systems of n real numbers x1, …xn. The number n is the dimension of the manifold. The spaces that are studied in most geometric theories are manifolds; the simplest geometric figures (segments, parts of surfaces bounded by curves, and so forth) are usually portions of manifolds. If, among all the systems of coordinates that can be introduced in portions of a manifold, we single out systems of coordinates that are linked by differentiable (a certain number of times) or analytic functions, then a so-called smooth (analytic) manifold is obtained. This concept generalizes the intuitive idea of a smooth surface. Smooth manifolds as such constitute the subject matter of differential topology. In geometry proper they are endowed with added properties. Coordinates whose transformations are supposed to be differentiable serve as the basis for the broad use of analytic methods—the differential and integral calculus as well as vector and tensor analysis. The aggregate of geometric theories developed by these methods forms general differential geometry. Its simplest instance is the classical theory of smooth curves and surfaces, which are just one- and two-dimensional differentiable manifolds.
(3) The generalization of the concept of motion as a transformation of one figure into another leads to the general principle of defining a space as a set of elements (points) with a given group of one-to-one transformations of this set onto itself. The “geometry” of such a space consists in studying those properties of figures that remain invariant under the transformations of this group. Therefore from the standpoint of such a geometry two figures can be considered “congruent” if one can be mapped onto the other by means of a transformation from the given group. For example, Euclidean geometry studies the properties of figures that remain invariant under motion; affine geometry studies the properties of figures that remain invariant under affine transformations; topology studies the properties of figures that remain invariant under one-to-one bicontinuous transformations. This pattern includes Lobachevskii’s geometry, projective geometry, and others. Actually, this principle is connected to the introduction of coordinates. A space is defined as a smooth manifold in which transformations are given by functions linking the coordinates of each given point and of the point to which it is being mapped (the coordinates of the image of a point are given as functions of the coordinates of the point itself and of the parameters on which the transformation depends; for example, affine transformations are defined by means of linear functions: x1’ = ai1x1 + ai2x2 + … + ainxn for i = 1,…, n). Therefore, the general apparatus for elaborating such “geometries” is the theory of continuous groups of transformations. Another, actually equivalent, viewpoint is possible, according to which it is not transformations of a space that are given but transformations of the coordinates in it, and what is studied are the properties of figures that are expressed identically in different systems of coordinates. This viewpoint found an application in the theory of relativity, which requires the identical expression of physical laws in different coordinate systems, which in physics are called frames of reference.
(4) Another general principle of defining spaces, pointed out in 1854 by Riemann, stems from a generalization of the concept of distance. According to Riemann, a space is a smooth manifold with a given law of measuring distances, or more accurately lengths, by infinitesimal steps. In other words, we are given the differential of the length of an arc of a curve as a function of the coordinates of a point on the curve and their differentials. This is a generalization of the intrinsic geometry of surfaces, which was defined by Gauss as a theory of the properties of surfaces that can be established by measuring the lengths of curves on them. The simplest case is that of so-called Riemannian spaces, in which the Pythagorean theorem is valid locally (that is, coordinates can be introduced in the neighborhood of every point so that at each point the square of the differential of the length of an arc will be equal to the sum of the squares of the differentials of the coordinates, and in arbitrary coordinates it is expressed in terms of a general positive definite quadratic form). Consequently, such a space is locally Euclidean but it may not be globally Euclidean, just as a curved surface can only locally be reduced to a plane (with appropriate precision). The geometries of Euclid and Lobachevskii prove to be a particular case of this Riemannian geometry. The broadest generalization of the concept of distance led to the concept of a general metric space as a set of elements in which a “metric” is defined, that is, a number is assigned to each pair of elements—the distance between them—and is subject only to very general conditions. This idea plays an important role in functional analysis and underlies several of the newest geometric theories, such as the intrinsic geometry of nonsmooth surfaces and the corresponding generalizations of Riemannian geometry.
(5) The linking of Riemann’s idea of defining the geometry of a manifold locally with the definition of “geometry” by means of a group of transformations led (E. Cartan, 1922-25) to the concept of a space in which transformations are assigned only locally. In other words, transformations here establish a connection only between infinitely close portions of a manifold: one portion is mapped into another, infinitely close one. Hence, one speaks of spaces with a “connection” of a certain type. Specifically, spaces with “Euclidean connection” are Riemannian. Further generalizations give rise to the concept of a space as a smooth manifold on which there is defined in general a “field” of some “object,” which can be a quadratic form, as in Riemannian geometry, an aggregate of quantities defining connection, a certain tensor, and so forth. One can also add here fiber spaces, which were introduced recently. These concepts include, in particular, a generalization of Riemannian geometry associated with the theory of relativity, where spaces are considered in which a metric is defined already not by means of a positive definite quadratic form but by means of a quadratic form with alternating signs (such spaces are also called Riemannian, or pseudo-Riemannian if one wishes to distinguish them from Riemannian in the original sense). These spaces are spaces with connection defined by the corresponding group, which differs from the group of Euclidean motions.
On the basis of the theory of relativity a theory of spaces arose in which the concept of order of points is defined such that to each point X there corresponds a set V(X) of points following it. (This is a natural mathematical generalization of a sequence of events, which is defined by the fact that event Y follows event X if X affects Y, and then Y follows X in time in any frame of reference.) Since the very assigning of sets V defines the points that follow X as belonging to the set V(X), the definition of this type of space proves to be an application of the first of the foregoing principles, where the “geometry” of a space is defined by distinguishing special sets. Of course, at the same time the sets V must obey the appropriate conditions; in the simplest case they are convex cones. This theory includes the theory of the corresponding pseudo-Riemannian spaces.
(6) The axiomatic method in its pure form now serves either for the formalization of developed theories or for defining general types of spaces by singling out special sets. If a certain type of more concrete spaces is defined by formulating their properties as axioms, then one uses coordinates, a metric, and so forth. The consistency and consequently the validity of an axiomatic theory are verified by constructing a model that realizes the theory, as was done for the first time for Lobachevskii’s geometry. The model itself is constructed from abstract mathematical objects, so the “final substantiation” of any geometric theory enters the domain of the foundations of mathematics, which cannot be final in the full sense but require more investigation.
The foregoing principles in different combinations and variations give rise to a vast diversity of geometric theories. The significance of each of them and the degree of attention to its problems are determined by the meaningfulness of these problems and of the results that are obtained and by its connections with other theories of geometry, with other fields of mathematics, with exact natural science and with problems of technology. A given geometric theory differs from other geometric theories in various ways. First, there is the space or type of spaces considered in it. Second, the definition of the theory includes a reference to the figures that are investigated in it. In this way a distinction is drawn between the theories of polyhedrons, curves, surfaces, convex bodies, and so forth. Each of these theories can be developed in one space or another. For example, it is possible to examine the theory of polyhedrons in ordinary Euclidean space, in an n-dimensional Euclidean space, in Lobachevskii space, and in other spaces. Similarly, it is possible to develop a theory of surfaces in Euclidean space, in projective theory, in a Lobachevskii space, and so forth. Third, the nature of the properties of figures that are being considered is of importance. For example, one can study the properties of surfaces that remain invariant under various transformations; one can consider the theory of the curvature of surfaces, the theory of bending (that is, of deformations that do not change the lengths of curves on a surface), and intrinsic geometry. Finally, the definition of a theory may include its basic method and the method of the formulation of problems. In this way, a distinction is drawn between elementary, analytic, and differential geometries; for example, one may speak of elementary or analytic geometry of Lobachevskii space. Local geometry, which examines only the properties of infinitesimal portions of a geometric object (curve, surface, manifold), is distinguished from global geometry, which studies geometric objects as a whole over their entire extent. We usually distinguish between analytic methods and the methods of synthetic geometry (or intrinsically geometric methods), the former make use of an appropriate calculus—differential, tensor, and others—while the latter operate directly with geometric objects.
Of the entire diversity of geometric theories, n-dimensional Euclidean geometry and Riemannian (including pseudo-Riemannian) geometry are in fact developing the most. In the former, the theory of curves and surfaces (and hypersurfaces of various dimensions) is intensively elaborated, with particular attention being given to the global investigation of surfaces and to the study of surfaces that are substantially more common than the smooth surfaces studied in classical differential geometry. This category includes polyhedrons (polyhedral surfaces). Then one must cite the theory of convex bodies, which, however, can be largely included in the global theory of surfaces, since a body is determined by its surface. Further, we must mention the theory of regular systems of figures, that is, those allowing motions that take the entire system to itself and some figure in it to any other one. It may be noted that a considerable number of highly important results in these fields belong to Soviet geometers: a full elaboration of the theory of convex surfaces and substantial development of the theory of general nonconvex surfaces, diverse theorems on surfaces in the large (the existence and uniqueness of convex surfaces with an assigned intrinsic metric or with a certain assigned “curvature function,” the theorem on the impossibility of the existence of a complete surface with a curvature that everywhere is less than some negative number, and others), investigation of regular division of space, and so forth.
The theory of Riemannian spaces investigates, among others, problems that concern the connection between their metric properties and topological structure; the global behavior of geodesies (locally shortest curves), for example, the question of the existence of closed geodesies; questions of imbedding, that is, the realization of a given m-dimensional Riemannian space in the form of an m-dimensional surface in Euclidean space of a certain number of dimensions; and questions of pseudo-Riemannian geometry associated with the general theory of relativity. To this one may add the development of various generalizations of Riemannian geometry, both in the spirit of general differential geometry and in the spirit of generalizations of synthetic geometry.
In addition, one should mention algebraic geometry, which developed from analytic geometry and investigates primarily geometric objects that are defined by algebraic equations. It occupies a special place, since it includes not only geometric but also algebraic and arithmetic problems. A vast and important field of research into infinite-dimensional spaces also exists, which, however, is not included under geometry but rather under functional analysis, since infinite-dimensional spaces are concretely defined as spaces whose points are certain functions. Nevertheless, there are many results and problems in this field that are authentically geometric in character and that therefore should be classified under geometry.
Significance of geometry. The use of Euclidean geometry is the most common phenomenon wherever areas, volumes, and the like are being determined. All technology, inasmuch as the forms and dimensions of objects play a role in it, makes use of Euclidean geometry. Cartography, geodesy, astronomy, all graphic methods, and mechanics are inconceivable without geometry. A prominent example is J. Kepler’s discovery that planets revolve along ellipses. He was able to take advantage of the fact that the ellipse had been studied already by ancient geometers. A profound application of geometry is geometric crystallography, which has served as a source and field of application for the theory of regular systems of figures.
More abstract geometric theories find wide use in mechanics and physics, when the totality of states of some system is viewed as a certain space (see above: Generalization of the subject matter of geometry). For instance, all the possible configurations (mutual disposition of elements) of a mechanical system form a “configurational space”; the motion of a system is represented by the motion of a point in this space. The totality of all states of a physical system (in the simplest case, the positions and velocities of material points forming the system, as for example, the molecules of a gas) is viewed as the “phase space” of the system. This point of view finds use, in particular, in statistical physics.
The concept of a multidimensional space was derived in connection with mechanics by J. Lagrange, when to the three spatial coordinates x, y, and z, a fourth was formally added—time t. In this way the four-dimensional space-time appeared, where a point is determined by the four coordinates x, y, z, and t. Every event is characterized by these four coordinates and, abstractly, the set of all events in the world proves to be a four-dimensional space. This view was developed in the geometric treatment of the theory of relativity given by H. Minkowski and later in A. Einstein’s construction of the general theory of relativity. In it he made use of a four-dimensional Riemannian (pseudo-Riemannian) geometry. Thus geometric theories that had developed from a generalization of the data of spatial experience proved to be a mathematical method for constructing a more profound theory of space and time. The theory of relativity, in turn, provided a powerful impetus for the development of general geometric theories. Initially arising from elementary practice, geometry returned, through a series of abstractions and generalizations, to natural science and practice at a higher level as a method.
From the geometric standpoint, the space-time manifold is usually treated in the general theory of relativity as an in-homogeneous manifold of the Riemannian type, but with a metric defined by a form with alternating signs that can be reduced locally to the formdx2 + dy2 + dz2 - c2dt2
where c is the velocity of light in a vacuum. The space itself, since it can be separated from time, also turns out to be inhomogeneous and Riemannian. From the standpoint of modern geometry, it is better to look at the theory of relativity as follows. The special theory of relativity states that a space-time manifold is a pseudo-Euclidean space, that is, one in which the role of “motions” is played by transformations that preserve the quadratic formx2 + y2 + z2 - c2t2
More precisely, this is a space with a group of transformations preserving the foregoing quadratic form. It is required of any formula expressing a physical law that it remain invariant under transformations of the group of this space, which are the so-called Lorentz transformations. But according to the general theory of relativity, the space-time manifold is inhomogeneous and is only locally pseudo-Euclidean, that is, it is a Cartan space (see above: Modern geometry). However, this insight became possible only later, since the concept of such spaces appeared after the theory of relativity and was developed under its direct influence.
In mathematics itself the position and role of geometry are determined above all by the fact that continuity was introduced into mathematics through geometry. Mathematics as a science of forms of reality encounters above all two general forms: discreteness and continuity. The counting of separate (discrete) objects yields arithmetic, while spatial continuity is studied by geometry. One of the major contradictions that stimulates the development of mathematics is the clash between the discrete and the continuous. The division of continuous quantities into parts and measurement already represent a juxtaposition of the discrete and the continuous— for example, a scale is laid out along a measured segment in separate steps. The conflict emerged with particular clarity when in ancient Greece (probably in the fifth century B.C.) the incommensurability of the side and diagonal of a square was discovered: the length of the diagonal of a square with side 1 was not expressible by any number, since the concept of an irrational number did not exist. It was necessary to generalize the concept of a number and create the concept of an irrational number (which was done only much later in India). The general theory of irrational numbers was created only in the 1870’s. A straight line (and together with it any figure as well) began to be viewed as a set of points. Now this point of view is the predominant one. However, the difficulties of set theory showed its limitations. The conflict between the discrete and the continuous cannot be totally resolved.
The general role of geometry in mathematics also consists in its connection with the precise synthetic thinking proceeding from spatial concepts, which often makes it possible to encompass as a whole that which is achieved by analysis and calculations only through a long series of steps. For instance, geometry is characterized not only by its subject matter but also by its method, which proceeds from visual representations and proves fruitful in solving many problems in other fields of mathematics. Geometry, in turn, makes wide use of their methods. Thus one and the same mathematical problem can quite often be treated either analytically or geometrically or by a combination of both methods.
In a certain sense, almost all mathematics may be viewed as developing from the interaction between algebra (originally arithmetic) and geometry, and in terms of method, as developing from a combination of calculations and geometric representations. This is evident already in the concept of the aggregate of all real numbers as the number line, which links the arithmetic properties of numbers with continuity. Here are several major aspects of geometry’s influence in mathematics.
(1) Geometry, along with mechanics, was of decisive importance in the origin and development of analysis. Integration originated from the determination of areas and volumes, begun already by the ancient scholars; moreover, area and volume as quantities were considered definite. No analytic definition of an integral was given until the first half of the 19th century. The drawing of tangents was one of the problems that gave rise to differentiation. The graphic representation of functions played an important role in developing concepts of analysis and retains its value. In the very terminology of analysis the geometric source of its concepts is evident, as for example, in the terms “point of discontinuity” and “domain of a variable.” The first course in analysis, written in 1696 by G. l’Höpital, was called Infinitesimals Analysis for the Comprehension of Curves. Most of the theory of differential equations is treated geometrically (for example, integral curves). The calculus of variations arose and is developing in large part from the problems of geometry, and the concepts of geometry play an important role in the calculus of variations.
(2) Complex numbers finally became established in mathematics at the turn of the 19th century only as a result of identifying them with the points of a plane, that is, by constructing the “complex plane.” A substantial role is allotted to geometric methods in the theory of functions of a complex variable. The very concept of an analytic function w =f(z) of a complex variable can be defined purely geometrically: such a function is a conformal mapping of a z-plane (or of a domain of a z-plane) to a w-plane. The concepts and methods of Riemannian geometry find an application in the theory of functions of several complex variables.
(3) The basic idea of functional analysis is that functions of a given class (for example, all continuous functions defined on a segment [0, 1]) are viewed as points of a “function space,” and the relationships between the functions are interpreted as geometric relationships between the corresponding points (for example, the convergence of functions is treated as the convergence of points and the maximum of the absolute value of the differences between two functions is defined as their distance. Then, many questions of analysis receive geometric elucidation, which in many cases proves to be very fruitful. In general, the representation of various mathematical items (such as functions and figures) as points of some space with the corresponding geometric interpretation of the relationships of these items is one of the most common and fruitful ideas of modern mathematics and has penetrated into almost all of its branches.
(4) Geometry exerts an influence on algebra and even on arithmetic (the theory of numbers). In algebra, for example, the concept of a vector space is used. In the theory of numbers a geometric trend has arisen that makes it possible to solve many problems that barely lend themselves to the computational method. By the same token, one must also take note of the graphic methods of calculations and the geometric methods of the modern theory of computations and computers.
(5) The logical perfecting and analysis of the axiomatics of geometry played a determining role in the development of an abstract form of the axiomatic method with its complete abstraction from the nature of objects and relationships that appear in the axiomatized theory. The concepts of the consistency, completeness, and independence of axioms were developed using the same material.
On the whole, the inter penetration of geometry and other fields of mathematics is so complete that often the boundaries prove to be matters of convention and tradition. Only such divisions as abstract algebra and mathematical logic have practically no connection with geometry.
REFERENCESBasic classical works
Euclid. Nachala, books 1-15. Moscow-Leningrad, 1948-50. (Translated from Greek.)
Descartes, R. Geometriia. Moscow-Leningrad, 1938. (Translated from Latin.)
Monge, G. Prilozheniia analiza k geometrii. Moscow-Leningrad, 1936. (Translated from French.)
Poncelet, J. V. Traité des propriét’s projectives des figures. Metz-Paris, 1822.
Gauss, K. F. “Obshchie issledovaniia o krivykh poverkhnostiakh.” In the collection Ob osnovaniiakh geometrii. Moscow, 1956. (Translated from German.)
Lobachevskii, N. l.Poln. sobr. soch, vols. 1-3. Moscow-Leningrad, 1946-51.
Bolyai, J. Appendix. Prilozhenie, … Moscow-Leningrad, 1950. (Translated from Latin.)
Riemann, B. “O gipotezakh, lezhashchikh v osnovaniiakh geometrii.” In the collection Ob osnovaniiakh geometrii. Moscow, 1956. (Translated from German.)
Klein, F. “Sravnitel’noe obozrenie noveishikh geometricheskikh issledovanii (’Erlangenskaia programma’).” Ibid.
Cartan, E. “Gruppy golonomii obobshchennykh prostranstv.” In the book VIII-i Mezhdunarodnyi konkurs na soiskanie premii imeni Nikolaia Ivanovicha Lobachevskogo (1937 god). Kazan, 1940. (Translated from French.)
Hilbert, D. Osnovaniia geometrii. Moscow-Leningrad, 1948. (Translated from German.)
Kol’man, E. Istoriia matematiki v drevnosti. Moscow, 1961.
Iushkevich, A. P. Istoriia matematiki v srednie veka. Moscow, 1961.
Wieleitner, G. Istoriia matematiki ot Dekarta do serediny 19 stoletiia, 2nd ed. Moscow, 1966. (Translated from German.)
Cantor, M. Vorlesungen über die Geschichte der Mathematik, vols. 1-4. Leipzig, 1907-08.
COURSESFoundations of geometry
Kagan, V. F. Osnovaniia geometrii, Part 1. Moscow-Leningrad, 1949.
Efimov, N. V. Vysshaia geometriia, 4th ed. Moscow, 1961.
Pogorelov, A. V. Osnovaniia geometrii, 3rd ed. Moscow, 1968.
Hadamard, J. Elementarnaia geometriia. Part 1, 3rd ed., Moscow, 1948; part 2, Moscow 1938. (Translated from French.)
Pogorelov, A. V. Elementarnaia geometriia. Moscow, 1969.
Aleksandrov, P. S. Lektsii po analiticheskoi geometrii. … Moscow, 1968.
Pogorelov, A. V. Analiticheskaia geometrii, 3rd ed. Moscow, 1968.
Rashevskii, P. I. Kurs differentsial’noi geometrii, 3rd ed. Moscow-Leningrad, 1950.
Kagan, V. F. Osnovy teorii poverkhnostei v tenzornom izlozhenii, parts 1-2. Moscow-Leningrad, 1947-48.
Pogorelov, A. V. Differentsial’naia geometriia. Moscow, 1969.
Descriptive and projective geometry
Glagolev, N. A. Nachertatel’naia geometriia, 3rd ed. Moscow-Leningrad, 1953.
Efimov, N. V. Vysshaia geometriia, 4th ed. Moscow, 1961.
Riemannian geometry and its generalizations
Rashevskii, P. K. Rimanova geometriia i tenzornyi analiz, 2nd ed. Moscow-Leningrad, 1964.
Norden, A. P. Prostranstva affinoi sviaznosti. Moscow-Leningrad, 1950.
Cartan, E. Geometriia rimanovykh prostranstv. Moscow-Leningrad, 1936. (Translated from French.)
Eisenhart, L. P. Rimanova geometriia. Moscow, 1948. (Translated from English.)
Some monographs on geometry
Fedorov, E. S. Simmetriia i struktura kristallov. Osnovnye raboty. Moscow, 1949.
Aleksandrov, A. D. Vypuklye mnogogranniki. Moscow-Leningrad, 1950.
Aleksandrov, A. D. Vnutrenniaia geometriia vypuklykh poverkhnostei. Moscow-Leningrad, 1948.
Pogorelov, A. V. Vneshniaia geometriia vypuklykh poverkhnostei. Moscow, 1969.
Busemann, H. Geometriia geodezicheskikh. Moscow, 1962. (Translated from English.)
Busemann, H. Vypuklye poverkhnosti. Moscow, 1964. (Translated from English.)
Cartan, E. Metod podvizhnogo repera, teoriia nepreryvnykh grupp i obobshchennye prostranstva. Moscow-Leningrad, 1936. (Translated from French.)
Finikov, S. P. Metod vneshnikh form Kartana v differentsial’noi geometrii. Moscow-Leningrad, 1948.
Finikov, S. P. Proektivno-differentsial’naia geometriia. Moscow-Leningrad, 1937.
Finikov, S. P. Teoriia kongruentsii. Moscow-Leningrad, 1950.
Schouten, J. A., and D. J. Struik. Vvedenie v novye metody differentsial’noi geometrii, vols. 1-2. Moscow-Leningrad, 1939-48. (Translated from English.)
Nomizu, K. Gruppy Li i differentsial’naia geometriia. Moscow, 1960. (Translated from English.)
Milnor, J. Teoriia Morsa. Moscow, 1965. (Translated from English.)
A. D. ALEKSANDROV