# embedding

(redirected from Isometric immersion)
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## embedding

[em′bed·iŋ]
(mathematics)
An injective homomorphism between two algebraic systems of the same type.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## embedding

(mathematics)
One instance of some mathematical object contained with in another instance, e.g. a group which is a subgroup.

## embedding

(theory)
(domain theory) A complete partial order F in [X -> Y] is an embedding if

(1) For all x1, x2 in X, x1 <= x2 <=> F x1 <= F x2 and

(2) For all y in Y, x | F x <= y is directed.

("<=" is written in LaTeX as \sqsubseteq).
References in periodicals archive ?
There exists an isometric immersion of ([M.sup.2], g) into [M.sup.2] (c) x [R.sup.2] (resp.
Roth, Isometric immersion into Lorentzian products, Int.
The case c = 0, that is, for isometric immersions in [R.sup.4], has been treated in .
Then f is a minimal isometric immersion. Consequently, there are no pseudohermitian immersions from a compact oriented strictly pseudoconvex CR manifold of CR dimension n into the Heisenberg group [H.sub.n+k] (carrying the standard strictly pseudoconvex pseudohermitian structure).
Here [Beta](f) denotes the second fundamental form (of the isometric immersion f) while [Mathematical Expression Omitted] is the Levi-Civita connection of (M, [g.sub.[Theta]]).
We wish to obtain CR analogues of the imbedding (Gauss-Ricci-Codazzi) equations of an isometric immersion. Let f : M [approaches] A be a pseudohermitian immersion.
Similarly, f* will not be related to an isometric immersion in Example 2.
By using (1), it can be easily shown that an isometric immersion f given in Proposition 2 is related to one in Example 1 if and only if for a constant [theta], (--[pi] < 2[theta] [less than or equal to] [pi]), the following equations hold identically.
If s = 0 in Lemma 1, the functions [u.sub.1] and [u.sub.2] are identically zero, and hence f is a standard isometric imbedding of [R.sup.2] into [R.sup.4] with a standard basis {[e.sub.1],[e.sub.2],[e.sub.3],[e.sub.4]} along the isometric immersion f.
Vlachos: Isometric immersions of warped products, arXiv:1108.3905, 2012.
Nolker: Isometric immersions of warped products, Differential Geom.
Stiel, Isometric immersions of constant curvature manifolds, Michigan Math.

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