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In an effort to understand some geometric rigidity theorems, I am curious about the following: let $(M,g)$ be a complete Riemannian manifold and suppose there is a nonconstant real-valued function on …

This is the differential form of the Reilly formula. It holds for a function on any pseudo-Riemannian manifold. (Robert C. Reilly. Applications of the Hessian operator in a Riemannian manifold, Indian …

Given any hypersurface of a Riemannian manifold with a unit normal vector field $\nu$, extend $\nu$ to be unit length. Then $$\operatorname{div}(u\nu)=u\operatorname{div}\nu+\text{d}u(\nu).$$ This pro …

Is the answer to the following problem, or some close variant thereof, known? Briefly: Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime satisfyi …

Given a smooth pseudo-Riemannian manifold $(M,g)$ one can define the conformal group as the set of smooth diffeomorphisms $\varphi:M\to M$ such that there is a positive smooth function $u$ with $\varp …

Given a connection $A_{i\alpha}^\beta$ the curvature is $$F_{ij\alpha}^\beta=\frac{\partial A_{j\alpha}^\beta}{\partial x^i}-\frac{\partial A_{i\alpha}^\beta}{\partial x^j}+A_{i\gamma}^\beta A_{j\alph …

It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent space into th …

Isn't this the Myers-Steenrod theorem? "If $(M,g)$ and $(N,h)$ are connected Riemannian manifolds and $f:(M,d_g)\to(N,d_h)$ is an isometry, then $f:(M,g)\to(N,h)$ is a smooth isometry"

The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. Ther …

I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-Mil …

See Example 3.1 in volume 2 of Kobayashi–Nomizu; the first Chern class is represented by $-\frac{1}{2\pi i}K$. Tensor with the Kähler form $m-1$ times and integrate over $X$.

In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July 20 …

Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs: J.G. Wolfs …

Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel …

Let $h^{\overline{i}j}$ be the inverse components, so that $h^{\overline{i}j}h_{j\overline{k}}=\delta_{\overline{k}}^{\overline{i}}$. If $\alpha$ is a (2,1)-form then $|\alpha|^2=h^{\overline{i}j}h^{\ …