Consider a submanifold $\mathcal{M}$ that is embedded in a higher dimensional manifold $\mathcal{N}$. Now if I infinitesimally perturb the submanifold as $$g_{\alpha\beta}\rightarrow g_{\alpha\beta}+\varepsilon R_{\alpha\beta}+\mathcal{O}(\varepsilon^2)$$ then
a) Is there an embedding of the new manifold in $\mathcal{N}$? If yes, is the embedding also infiitesimally perturbed from the original one? (Of course an embedding may not be unique, I mean is there one embedding that is infinitesimally close?)
b) If the answer to 'a' is affirmative, can I describe the embedding flow solely in terms of the extrinsic curvature?
I have shown that except for manifolds with constant curvature ($R_{\alpha\beta}=Cg_{\alpha\beta}$), the Ricci flow is different from the 'curve shortening flow' and therefore do not know where the animations of the ricci flow come from!