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The Sasaki metric gives a natural way to equip $TM$ with a Riemannian metric in case $M$ is already equipped with a Riemanian metric. Question: Let $M$ be manifold equipped with an affine connection …

Let $\gamma_1,\gamma_2:\mathbb{R}\rightarrow\mathbb{R}^n$ ($n\geq 1$) be two smooth curves such that for every $\,t_2,t_1\in\mathbb{R}$ we have $|\gamma_1(t_2)-\gamma_1(t_1)|=|\gamma_2(t_2)-\gamma_2 …

Set $M=\{(\cos(\theta),\sin(\theta),z):\theta\in[0,2\pi],z\in[0,1]\}$. A bending of $M$ is a smooth map $\Gamma:M\times [0,1]\rightarrow \mathbb{R}^3$ such that 1) $\Gamma[M\times\{t\}]$ is a submani …

Let $M$ be a connected $n$ dimensional boundary-less smooth manifold with the property that for any connected boundary-less $n$ dimensional manifold $\overline{M}$ and any embedding $i:M\rightarrow \o …

Let $M_1,M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M_1)\cong \mathfrak{X}(M_2)$ as Lie algebras. Questio …

Let $f:M\rightarrow N$ be a smooth map between two simply connected Riemannian manifolds of the same dimension. It is also given that for every $x\in M$ we have that $Df|_x:T_xM\rightarrow T_{f(x)}N$ …

Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $S$ be the vector space of multilinear maps from $V\times V$ to $V$. Let $L:\mathbb{I}\rightarrow S$ be a continuous map such that f …

Bellows conjecture for polyhedra was setteled in 1997. How about the smooth version of it, ie bending of closed 2D submanifolds in $\mathbb{R}^3$ while preserving the Riemannian structure/intrinsic ge …