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I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly: What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin …

I'll add a few things which I've found helpful to get intuition to what Renato's already written. Scalar curvature has the very simple geometric interpretation as the volume defect of small balls (as …

Is there a $C^\infty$-smooth embedding $\gamma : I \to \mathbb{R}^3$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?

Perhaps a more geometric way of viewing mean curvature (at least for a hypersurface) is as follows: If $\varphi: \Sigma^n\hookrightarrow (M^{n+1},g)$ is an embedded (oriented) hypersurface, then we …

In dimension $n\geq 3$, there cannot be any sort of bound on the multiplicities which does not depend on some geometric input. This is because it is a theorem of Colin de Verdière (mathscinet and arti …

An interesting "curvature" which has recently received much interest is the "Ma--Trudinger--Wang" (MTW) tensor, which arose in the study of when optimal transport maps are smooth on a Riemannian manif …

Is the converse to the Bishop-Gromov Inequality true? In other words, if, for a complete $n$-dimensional Riemannian manifold $M$, there is $k \in \mathbb{R}$, such that defining $V_k(R)$ to be the s …

The answer is yes (I'm assuming you are asking about closed manifolds, non-compactness allows for all sort of crazy things to happen, you can check out the work of Topping and collaborators). Kotsch …

Yang and Yau proved that for a surface of genus $\gamma$, $\Sigma$ with a metric $g$, the first eigenvalue satisfies $$ \lambda_1(g) Area(g) \leq 8\pi (1+\gamma). $$ So, the answer to your first ques …

Without embeddedness, the Choi--Schoen theorem is false. For example, there is a huge family of rotationally symmetric immersed tori in $\mathbb{S}^3$ (the only embedded one is the Clifford torus, b …

For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B_1|$, let $$ C(\Omega) = \int_\Omega\int_\Omega \frac{dxdy}{|x-y|} $$ denote the Coulomb (or gravitational, etc) energy. Poincaré is cred …

Misha's comment could be a bit misleading. In particular, it is not true that the Ricci flow should exist on a slightly bigger interval $(-\epsilon,T)$ with $g(0) = g_0$. One way to see this is by thi …

In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark here (page 246) …

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I …

Your claim that "all solutions from general relativity are geodesically incomplete" is not true. The classical Schwarzschild/Kerr black hole solutions are geodesically incomplete, and along with Minko …