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It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that $C^3$-isometries preserve the sectional curvature and for the surface case, it has been shown by Hartman and Wintner in "On the Fundamental Equations of Differential Geometry", that the statement remains true for $C^2$-isometries. Does someone know a reference for general dimension? It seems likely that $C^2$ is enough.

Thanks for clarification.

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A $C^1$-isometry of a $C^\infty$ Riemannian manifold is already $C^\infty$, by elliptic regularity, for example. The group of all isometries is always a finite dimensional Lie group.

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  • $\begingroup$ 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? $\endgroup$
    – frog
    Commented May 12, 2014 at 10:02
  • $\begingroup$ 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 $N$. $\endgroup$
    – Deane Yang
    Commented May 12, 2014 at 16:13
  • $\begingroup$ 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. $\endgroup$
    – frog
    Commented May 13, 2014 at 6:53
  • $\begingroup$ 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? $\endgroup$
    – frog
    Commented May 13, 2014 at 6:55

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