# Tagged Questions

**0**

votes

**0**answers

118 views

### How to prove this inequality of heat flow from Weitzenbock formula?

Let $(M,g), (N,h)$be a compact Riemannian manifolds, $m:=\dim M, n:=\dim N\geq 2$,
and $N$ is a non-positive curvature $K_N\leq 0$. All connections which appear below are the Levi-Civita connections. ...

**2**

votes

**0**answers

63 views

### Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious.
General gist of the problem
I have a variational problem on a ...

**11**

votes

**2**answers

526 views

### A riemannian manifold with finitely many closed contractible geodesics

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if ...

**3**

votes

**0**answers

179 views

### Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds

I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...

**9**

votes

**2**answers

704 views

### Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...

**1**

vote

**1**answer

463 views

### Are geodesics locally minimizing in continuous curves?

In every lecture on Riemannian geometry it is standard to prove that geodesic curves are locally length minimizing.
The only thing I find confusing about this is, that here length minimizing means: ...

**7**

votes

**1**answer

701 views

### Calculating the geodesic equation for a particular set of phase-space coordinates

Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...

**1**

vote

**1**answer

617 views

### “Synthetic” proof of geodesic flow equation?

First, let me explain what I mean by "synthetic" in the title, which is a proof that reasons purely axiomatically and does not explicitly invoke local coordinate charts (either via concrete expansions ...