Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with riemannian-geometry
Search options not deleted
user 16702
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.
3
votes
Calculation of the top Chern class of spinor bundle over $S^{2n}$
I will present an explicit calculation using Chern-Weil theory, which makes an amusing use of Legendre's duplication formula for the gamma function.
The Chern character form of a vector bundle $E$ wit …
- 9,853
3
votes
2
answers
361
views
Exponential decay of resolvent kernel
For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function …
- 9,853
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 …
- 9,853
4
votes
Accepted
heat kernel on closed manifolds - error in Chavel's book?
Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that
$$L(H_k * F) = (LH_k)*F - F,$$
so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then
$$ L p = LH_ …
- 9,853
5
votes
Accepted
Stochastic interpretation of heat kernel on fiber bundle
Let $P\longrightarrow M$ be a $G$ principal bundle endowed with a connection $1$-form $\omega$ (which has values in the Lie algebra $\mathfrak{g}$). If $X_t^x$ denotes Brownian motion on $M$ starting …
- 9,853
3
votes
1
answer
338
views
Certain construction of the Itô integral on manifolds
Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, …
- 9,853
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 …
- 9,853
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
$$\nabla^* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\ …
- 9,853
3
votes
The complex heat kernel on a Riemann manifold
As far as I know, the term "Mehler Kernel" is used for the integral kernel of the heat equation corresponding the the harmonic oscillator,
$$ \partial_tu + \Delta u + x^2 u = 0.$$
The equation you are …
- 9,853
4
votes
1
answer
679
views
Horizontal lift of differential operator
On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a horizon …
- 9,853
7
votes
1
answer
894
views
Sharp Gaussian upper bounds on Heat Kernel
I am looking for references (with proof) for the following statement:
Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compac …
- 9,853
3
votes
de Rahm Laplace operator on forms bounded
Differential operators are never bounded unless they are of order zero.
Standard references are Berline, Getzler, Vergne, "Heat Kernels and Dirac Operators" and Gilkey, "Invariance Theory, The Heat E …
- 9,853
0
votes
Geodesic equation from Christoffel symbols
I do not see why your metric is a sensible object, because it depends highly on your choice of coordinates (I am not even sure that your formula defines a metric, but I may be overlooking something). …
- 9,853
2
votes
0
answers
209
views
Bounds on functions pullbacked via exponential map
Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* …
- 9,853
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 …
- 9,853