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Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$. How big can the set $\math …

In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else. We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal i …

I am an analyst struggling through some geometry used in physics. Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection …

I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it …

Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it …