All Questions

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$. Other than when $M$ is non-positively curved (in which $C_p= \emptyset$ by ...

$g_{ij}(x)\in L^\infty(\mathbb{R}^n, M^{n\times n})$ is a metric in $\mathbb{R}^n$ satisfying $\lambda |x|^2\leq g_{ij}x^ix^j\leq \Lambda |x|^2$($\lambda>0$&$\Lambda>0$) Can we find a ...

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...

The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere. I am wondering if this result can be generalized as follows. ...

Hallo, I have the following question: Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\...