All Questions

Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set $$ S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}. $$ Is correct to say that $S(...

Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...

A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...