Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent …

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel tra …

How can I compute the curvature of the contour lines (equipotential lines) $\phi (\vec{r})=c$ for the scalar field $\phi (\vec{r})$ ? I expect the direction of the curvature vector …

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vani …

In first course differential geometry you learn, that Ricci-curvature is something like a mean-value of the curvature endomorphism, because it's a trace, and the scalar curvature i …

Thought Experiment Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator. Case 1 Let us parallel transport a vector, $V$ from $p$ using the recipe: Move one unit o …

First let me state a result of Kazdan and Warner Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that has the same sig …

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, tha …

What are some open problems in sub-Riemannian geometry? I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.

Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of …

It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness …

Dear all, When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields $[\nabla_\mu,\nab …

I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider …

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the firs …

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits …