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Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

1 vote
0 answers
128 views

Volume growth of balls implies volume growth of spheres?

Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies $$\mathrm{vol}(B_R(p)) \leq v(R)$$ for some function $v$. Can we t …
Matthias Ludewig's user avatar
1 vote
2 answers
282 views

Number of geodesics of certain length

Let $M$ be a Riemannian manifold, and let $x, y \in M$ be non-conjugate points. Let $r, R>0$ be two numbers. I am looking for a bound on the number of geodesics between $x$ and $y$ of Length between …
Matthias Ludewig's user avatar
6 votes
0 answers
2k views

Second derivative of Riemannian Exponential Map

Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$. The derivative of the exponential map can be expressed in T …
Matthias Ludewig's user avatar
24 votes
7 answers
5k views

Difference between parallel transport and derivative of the exponential map

This is a crosspost from math.stackexchange Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then w …
Matthias Ludewig's user avatar
0 votes

Geodesic equation from Christoffel symbols

I do not see why your metric is a sensible object, because it depends highly on your choice of coordinates (I am not even sure that your formula defines a metric, but I may be overlooking something). …
Matthias Ludewig's user avatar
10 votes
Accepted

Relationship between Laplacian and Hessian on compact Lie groups

This has nothing to do with Lie groups, I believe. Let $M$ be a Riemannian manifold. The Bochner formula on $1$-forms states that $$\nabla^* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\ …
Matthias Ludewig's user avatar
2 votes
0 answers
104 views

Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy $$E[ …
Matthias Ludewig's user avatar
1 vote

How to define the square root of $1-\Delta $?

You are done once you know that you have a functional calulus for the Laplace-Beltrami operator on $M$. For this, show that it is self-adjoint and has nonpositive spectrum (there are various ways to d …
Matthias Ludewig's user avatar
3 votes
Accepted

Existence of Geodesics in continuous metrics

Ok, thank you Misha for the comments, let me try to fill out the hints you gave myself. I try to prove the following: Let $g_n$ be a sequence of complete smooth metrics that converge in $C^0$ agains …
Matthias Ludewig's user avatar
6 votes
1 answer
293 views

Heat Kernel Asymptotics with low regularity

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 …
Matthias Ludewig's user avatar
5 votes
1 answer
1k 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 g …
Matthias Ludewig's user avatar
7 votes
1 answer
894 views

Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compac …
Matthias Ludewig's user avatar
3 votes
0 answers
406 views

Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly b …
Matthias Ludewig's user avatar
3 votes
1 answer
307 views

Pullback of $L^p$ functions via exponential map

Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback $$ \exp^* u = u \circ \exp$$ which is in $C^k …
Matthias Ludewig's user avatar
5 votes
2 answers
3k views

Van Vleck-Morette Determinant

There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source an …
Matthias Ludewig's user avatar

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