Isometric $C^1$-immersions for pairs of Riemannian metrics

Partial Isometries of a Sub-Riemannian Manifold

“…The conditions (1) and (2) together guarantee that the sequence {f n } is a Cauchy sequence in the fine C 1 topology and hence it converges to some C 1 map f : M −→ N . Then the induced metric f * h must be equal to g H when restricted to H by condition (1). Thus f is the desired partial isometry.…”

“…Then g M = (g H − f * 0 h| H ) ⊕ g K + f * 0 h is a Riemannian metric on T M which clearly restricts to g H on H. Moreover, g M − f * 0 h > 0 on M . Then by Nash's decomposition formula (see [1] and [9]), there exist smooth functions ψ i and φ i as described in the lemma such that g M − f * 0 h = i φ 2 i dψ 2 i . By restricting both sides to H we get the desired decomposition.…”

“…M . Then by Nash's decomposition formula (see [1] and [9]), there exist smooth functions ψ i and φ i as described in the lemma such that g M − f * 0 h = i φ 2 i dψ 2 i . By restricting both sides to H we get the desired decomposition.…”

“…Suppose we have obtained fk satisfying (1)-( 4)at the end of the k-th step, where we can write f * k h| H = ḡk + δk . In the next step we want to increment the induced metric on H by a quantity φ 2 k+1 dψ 2 k+1 | H , that is, we want to induce ḡk+1 + δk on H. Taking g = ḡk+1 + δk and f = fk in Lemma 4.1 we obtain a smooth map fk+1 which clearly satisfies (1). If we write f * k+1 h = ḡk+1 + δk + δ k+1 , then δk+1 = k+1 i=1 δ i and hence (2) and (3) are clearly satisfied.…”

“…Consider the equation X(f 0 + α), X(f 0 + α) = 1, where α : M → R 2 is a smooth map. This reduces to (1) Xα, Xα + 2 Xf 0 , Xα = 1 − φ 2 .…”