Editing Embedding (section)

===Riemannian and pseudo-Riemannian geometry===

In [[Riemannian geometry]] and pseudo-Riemannian geometry:
Let <math>(M,g)</math> and <math>(N,h)</math> be [[Riemannian manifold]]s or more generally [[pseudo-Riemannian manifold]]s.
An '''isometric embedding''' is a smooth embedding <math>f:M\rightarrow N</math> that preserves the  (pseudo-)[[Riemannian metric|metric]] in the sense that <math>g</math> is equal to the [[pullback (differential geometry)|pullback]] of <math>h</math> by <math>f</math>, i.e. <math>g=f^{*}h</math>. Explicitly, for any two tangent vectors <math>v,w\in T_x(M)</math> we have

:<math>g(v,w)=h(df(v),df(w)).</math>

Analogously, '''isometric immersion''' is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of [[curve]]s (cf. [[Nash embedding theorem]]).<ref>Nash J.,  ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), '''63''' (1956), 20–63.</ref>