Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ …

On a compact Riemannian oriented manifold $M$,for each singular $k$-chain $\sigma$ (with real coefficients), $\sigma$ induces a linear functional on the $\mathbb{R}$-vector space o …

Here is an updated formulation of the question, which is more precise and I think completely correct: Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be …

Let $K(r)$ be the piecewise function                        &nbs …

It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle. What i …

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed …

What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"? I get the feeling that it is a choice of basis with non-trivial structure cons …

Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature …

In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system. Now if on the same space one has two such metrics gi …

I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry. Jurgen Jost's book does give somewh …

Given an isometric (in the Riemannian way) immersion $f:N\rightarrow M$ between complete, smooth riemannian manifolds, are there conditions on $M$, $N$, $f$, such that the normal e …

Consider a "bump surface" which looks like the following: Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature alo …

Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space. Then how are the Killing fields on $G/H$ the projection of the right-inv …

Wikipedia tells me that: Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane: $K = \lim_{r \rightarrow 0} (2 \p …

Given four non-planar 3D points, these points define a skew quadrilateral ABCD. Example: A = (0,0,0); B = (0,3,0); C = (3,3,2); D = (3,0,-1); The bilinear interpolation of …