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I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem: Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into 1 …

If you are willing to assume that the embedded surface $S$ is polyhedral, you can prove that it is orientable by an elementary argument similar to the proof of polygonal Jordan Theorem. Of course the …

I found that I need to use the following facts in a paper that I am writing. Let $f\in C^\infty(\mathbb R)$, then If $f(0)=0$, then $f(x)=x g(x)$ for some $g\in C^\infty(\mathbb R)$. If $f$ is even …

The statement is true. It is almost precisely Lemma 2 in the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210. The proof is short but not easy to invent. The paper can be read …

Upper bound (assuming the manifolds are second countable): every manifold admits a complete metric, and the "set" of isometry classes of complete separable metric spaces is of cardinality continuum. I …

I'm sure this is standard but I don't know where to look. Let $M$ be a contractible compact smooth $n$-manifold with boundary. Does it have to be homeomorphic to $D^n$? What about diffeomorphic? [UPD …

Q3: Laman's theorem is the same on the sphere. Indeed, a configuration with $n$ vertices and $m$ edges is defined by a system of $m$ equations in $2n-3$ variables (there are $2n$ coordinates of point …

In addition to Anton Petrunin's answer, here is a trick to simplify (and in some sense solve) the geodesic equation. Since the metric has three-dimensional group of isometries (generated by rigid moti …

Consider the following differential $(n-1)$-form $\omega$ on $M$: for $p\in M$ and $v_1,\dots,v_{n-1}\in T_pM$, define $$ \omega(v_1,\dots,v_{n-1}) = [ \psi(p), \nu(p), d\psi(v_1),\dots,d\psi(v_{n-1} …

I assume that by "all derivatives" you mean derivatives of every order. Suppose that all transitions between normal coordinates have uniformly bounded derivatives within some radius $r$. For every po …

I don't know a reference but maybe the following proof is shorter than yours. By continuity, $\varphi\circ\varphi_\infty=\varphi_\infty$. Hence $\varphi$ is identity on the set $I:=\varphi_\infty([0, …

I think what you are looking for is boundary at infinity. For example, the boundary at infinity of the hyperbolic plane $\mathbb H^2$ is its "ideal boundary" circle, and adding it to $\mathbb H^2$ yie …

No you can't have uniformly bounded second derivatives and diameters at the same time (unless you are allowed to change the Riemannian metric). Consider a smoothened boundary of a tubular neighborhoo …

It is not compact unless you allow the origin at the boundary (or maybe impose some kind of uniform strict convexity). Consider a rectangle $K=[-1,1]\times[-\delta,1]$ in the plane, where $\delta$ is …

I am not sure what the question is, but will try to answer anyway. This is basic general topology stuff, sorry if you meant something deeper in your question. For any topological space $X$ and any eq …