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This fact might be either trivial, wrong, or well known. Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy $$(m-u_1) \dots (m-u_{s-1})=0 …

Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact sup …

Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorna …

Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume $$II_{\partial M} …

I was told that the following facts are true. I am looking for a reference to them. 1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$. 2) The action of $SO(n,\m …

Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that $$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$ …

Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$. Question Is it true that there exists a continuous finite dimensional representation $ …

Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$ be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation $$\partial_t \vec u(x,t)=\sum_{j=1} …

It is well known and easy to see (modulo standard basic facts) that any compact 1-dimensional Alexandrov space with curvature bounded from below is isometric either to a circle or to a segment. I am l …

In his book "Some non-linear problems in Riemannian geometry" T. Aubin states the following result (Theorem 3.56): Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order differential opera …

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)? If yes, a reference would be helpful. EDIT: If the quest …

The Cheeger-Gromov compactness theorem says the following. Let us fix $n\in \mathbb{N}$ and positive constants $K,D,v$. Let $\{(M_i^n,g_i)\}$ be a sequence of closed infinitely smooth $n$-dimensional …

I am not sure that this is a research level question. Remark 10.9.4 in the book "A course in metric geometry" by Burago, Burago, Ivanov claims the following. Let $X$ be a finite dimensional Alexandr …

Let $X$ be an $n$-dimensional compact Alexandrov space with curvature bounded below which has non-empty boundary. Is it true that the boundary has Hausdorff dimension $n-1$? If yes, does it have finit …

I am looking for a reference to the following notions and facts below which, I think, I can prove, but which might be known to experts. Let us fix a finite group $G$. Consider the class of all compact …