2
$\begingroup$

Let $M$ be a differentiable manifold of dimension $n>1$ and $g$ a flat Riemannian metric on $M$. Consider $f:M\rightarrow \mathbb{R}^+$ a continuous function which doesn't have continuous first derivatives and $g':=e^{2f}g$ a non-smooth Riemannian metric on $M$.

Does the notion of curvature (curvature tensor, sectional curvature) still make sense for $(M,g')$? I'm searching for a reference which covers this aspect of singular riemannian metrics.

In the smooth case there is a formula to compute the curvature tensor $R'$ of $g'$ with respect to the derivatives of $f$:

$$ \begin{align} R'(X,Y)Z &= g(\nabla_X \operatorname{grad} f,Z)Y - g(\nabla_Y \operatorname{grad} f,Z)X\\ &+ g(X,Z)\nabla_Y \operatorname{grad} f - g(Y,Z)\nabla_X \operatorname{grad} f\\ &+ (Y f)(Z f)X - (X f)(Z f)Y\\ &- g(\operatorname{grad} f, \operatorname{grad} f)[g(Y,Z)X - g(X,Z)Y]\\ &+ [(X f)g(Y,Z)-(Y f)g(X,Z)]\operatorname{grad} f \end{align} $$

can this be used or adapted in case $f$ isn't $C^1$?

Improve this question
$\endgroup$

3 Answers 3

Reset to default
4
$\begingroup$

First you have to understand what is curvature of singular metric.

I think the best possible answer in dimension 2 was given by Reshetnyak in and I suspect that the proofs work in higher dimensions. (Reshetnyak was interested in dimension 2 since in this case all reasonable metrics are conformally flat.)

Check "Two-dimensional manifolds of bounded curvature” by Reshetnyak.

Improve this answer
$\endgroup$
3
$\begingroup$

Under stronger regularity assumptions, an analog of the curvature exists in the weak sense, i.e., in the sense of generalized functions. The stronger regularity assumption is that the metric (in your case, the conformal coefficient) is of class $W^{1, \infty}$, that is, there exists the first derivative of the components $g_{ij}$ which are locally bounded functions. In this setup, the entries $R^{i}_{\ j k \ell}$ should be viewed as distributions, that is, for any smooth locally-supported function $\phi $ one can calculate $\int_{M} \phi R^{i}_{\ j k \ell} $.

See e.g. S. Mardare, On isometric immersions of a Riemannian space under weak regularity assumptions. C. R. Math. Acad. Sci. Paris 337 (2003), no. 12, 785–790, and S. Bandyopadhyay et al, On the equation $(Du)tHDu=G$. Nonlinear Anal. 214 (2022), Paper No. 112554 where it is proved that if the curvature vanishes as distribution, then there exists a ``smoother'' coordinate system such that the components of the metric are flat.

Improve this answer
$\endgroup$
0
$\begingroup$

When you lose the regularity , the situation has to be evaluated case by case , I don't think there are " general procedures " to operate ...

Improve this answer
$\endgroup$

You must log in to answer this question.