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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 $X$.

There are many non-equivalent formulations of the Schauder estimates; the version you are suggesting is the most common. For the version he needs, Yau gives a precise page & theorem reference (p.156, …

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 …

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 …

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 …

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 …

For any 1-form $\omega$ on a pseudo-Riemannian manifold, the product rule and commutation identity say that $$\omega^p\nabla_p\nabla_q\omega^q-\nabla^q(\omega^p\nabla_p\omega_q)=-\omega^p\omega^qR_{pq …

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 …

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 …

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 …

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 …

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 …

I agree with Alexandre Eremenko in that most can be hard to read. But I think you can get a lot out of trying to understand them, as long as you're willing to black-box certain parts which may be inac …

Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one c …

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^{\ …