Timeline for Optimal Regularity for Invariance of Curvature under Isometries

5 events

when toggle format	what		by	license	comment
May 13, 2014 at 6:55	comment	added	frog		Curvature preserved means if $(M,g)$ and $f(M),\widetilde g)$ (where $\widetilde g$ is the induced metric from ambient space) have the same sectional curvature respectively whether it makes sense to speak of sectional curvature?
May 13, 2014 at 6:53	comment	added	frog		Right, I mean an isometric embedding of $M$ into some (possibly) higher dimensional manifold $N$. I know that $C^1$-isometric embeddings as constructed by Nash enjoy more flexibility due to the lack of curvature. If for example we consider an isometric embedding $f\in C^k(M,\mathbb R^d)$ for $d$ sufficiently big, $f(M)$ will be in general a $C^k$-manifold and its induced metric will be of class $C^{k-1}$. So I wonder what happenes with $C^2$-embeddings, whether it makes sense to speak of curvature and if yes, if it is preserved.
May 12, 2014 at 16:13	comment	added	Deane Yang		What does "curvature preserved" mean for an isometric embedding? I'm assuming you mean the isometric embedding of a $M$ into a higher dimensional manifold $N$.
May 12, 2014 at 10:02	comment	added	frog		Thanks for your answer. What about embeddings? If $(M,g)$ is isometrically embedded in some other manifold $N$ by a $C^2$-isometry. Is curvature then preserved?
May 12, 2014 at 9:59	history	answered	Peter Michor	CC BY-SA 3.0