# Tagged Questions

A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

1k views

### Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
332 views

### Exponential map on the ellipsoid.

Consider the ellipsoid $M \subseteq \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{x^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian metric ...
205 views

132 views

### Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
417 views

### Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
896 views

### What is the volume of Complex Projective Space with Fubini-Study Metric?

I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
118 views

### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
43 views

### When a given family of curves are geodesics of some affine connection?

Let $M$ be a two-dimensional manifold and let $\mathcal C$ be a family of smooth paths on $M$. How to understand whether this family is actually a family of (possibly reparametrized) geodesics of some ...
151 views

### Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
118 views

### How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
118 views

### Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ (...