# Tagged Questions

**0**

votes

**0**answers

32 views

### Bounded Ricci curvature implies bound on Jacobi determinant?

Assume that $M$ is a complete Riemannian manifold and define for $X \in T_x M$
$$j(X) = \bigl|\det d \exp_x|_X \bigr|.$$
This is a smooth function on $TM$ for $X$ close enough to the zero section.
...

**2**

votes

**1**answer

36 views

### Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...

**2**

votes

**1**answer

371 views

### Existence of Geodesics in continuous metrics

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...

**2**

votes

**0**answers

148 views

### Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...

**7**

votes

**1**answer

646 views

### Is there an elementary way to show the triangular inequality for this expression ?

Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...

**5**

votes

**4**answers

1k views

### How does curvature change under perturbations of a Riemannian metric?

Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...

**1**

vote

**1**answer

364 views

### The comparison between the square of the functional value and the sum of squares of the L^2 norms of function and its Laplacian

I was reading a paper where I came across the following argument :
For any x in M and for a geodesic ball B(x; epsilon) in a compact Riemannian
manifold M with injectivity radius bigger than or ...

**3**

votes

**2**answers

553 views

### Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...