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This is a cross post of MSE post somehow: Is there any example of compact fiber bundle $E$ with noncompact fibers $F$? Obviously if the base space $B$ is $T_1$ then there is no such example.

I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)). …

Let $G$ be a simple graph with $n$ vertices. Let $\omega(G)$ and $\chi(G)$ denotes the clique number and chromatic number of $G$ respectively. Then Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for …

I want to know whether there are ways to use algebraic methods for solving graph theory problems (graph coloring problems). For example, is it possible to prove the four-color theorem purely with alg …

If I am not mistaken this has been done by R. Lakshminarayanan and you can find it in Bródy, F. (ed.); Vámos, T. (ed.), The Neumann compendium, World Scientific Series in 20th Century Mathematics. 1. …

This book must be useful: An Introduction to Differentiable Manifolds and Riemannian Geometry written by William Munger Boothby Page 319. Read online in Google Link.

Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$? …

This is a cross-post from MSE. These four screenshots from milnor's book baffled me a bit (pages 24, 50, 51 and i-iii resp.): In first one, there is no Theorem 3.1 in the book, but there i …

According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$. Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL …

This is a cross post of MSE. Q1: What does "irreducible manifold" mean (not definition)? My understanding of "irreducible manifold" is "is not reducible (homotopic or deformation or homeomorph or be …

Cross post from MSE. and sorry if this is an obvious question. Here is a line of proof of Theorem 1.15 from Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Prov …

In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference t …

For Q2 I found the following: In Ricci solitons and real hypersurfaces in a complex space form Cho and Kimura studied on Ricci solitons of real hypersurfaces in a non-flat complex space form and they …

I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions: $$\nabla \omega\neq 0,\quad \Delta\om …

Here is the comment to this book in author's web page: Differential Forms in Algebraic Topology (with Raoul Bott), third corrected printing, Graduate Text in Mathematics, Springer, New York, 1995. …