# umbilic

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## umbilic

[u̇m′bil·ik]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Then, p is an umbilic point if and only if rank[(df).sub.p] = 0 holds.
Since f has no umbilic point on U, the Gauss map v depends only on [u.sub.1] and[mathematical expression not reproducible] holds.
Three kinds of totally umbilic surfaces have H which are given by intersections of [H.sup.3] and hyperplanes HP(v, c) in [R.sup.4.sub.1] A surface HP(v, c) [intersection] [H.sup.3] is called H-sphere, equidistant surface, and horosphere if HP(v, c) is spacelike, timelike, and lightlike, respectively.
After the notion of congruent in [H.sup.3], we will give a different classification theorem of H-isometries in terms of leaving those totally umbilic surfaces of [H.sup.3] fixed.
Supposing M = X(U) is totally umbilic, we have the following propositions by simple computation.
Let [[lambda].sub.0] [member of] [R.sup.n + 1.sub.2] and let M be a Lorentzian surface without any umbilic point satisfying Kp(u) # 0.
Danko Arlington cast the Umbilic Torus SC (Simons Center) sculpture entirely out of silicon bronze, and the finished piece was welded together as an assembly at neighboring New Arts Foundry, a Baltimore shop that specializes in investment casting bronze.
If the curvature diagram is contained in the main diagonal [k.sub.1] = [k.sub.2] then the surface is a piece of a plane or a sphere because every point is an umbilic. The curvature diagram is contained in a straight line parallel to the diagonal [k.sub.1] = -[k.sub.2] if and only if the mean curvature is constant.
Aazami was informally testing out a special case of their evolving caustics theorem called an "ellyptic umbilic" by using a technical computing software program when he noticed a pattern.
KLEIN, On universality of critical behaviour in the focusing nonlinear Schrodinger equation, elliptic umbilic catastrophe and the tritronquee solution to the Painleve-I equation, J.
Moreover, owing to spherical symmetry, all points in the corresponding surfaces [summation] and [summation]' have constant Gaussian curvature relevant to their respective manifolds and centres of symmetry, so that all points in the respective surfaces are umbilic.
The spaces of the immediate future will be characterized by "complexity." They will derive from "catastrophe theory" and its language of structural stability, phase transitions, and continuous change, and with its basic types of discontinuities in the form of the fold, cusp, swallowtail, butterfly, and umbilic hyperbolic, elliptic, or parabolic.

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