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Results tagged with riemannian-geometry
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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
7
votes
2
answers
799
views
Asymptotic expansion of the Schrödinger kernel?
My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!
Let $M$ be a compact Riemannian manifold and $\Delta$ be the …
- 63.9k
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 …
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 …
- 637
5
votes
2
answers
2k
views
Triangle area on surfaces of constant curvature
I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the …
- 91.8k
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 …
- 356
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
$$ L p = LH_ …
- 9,853
5
votes
Accepted
Stochastic interpretation of heat kernel on fiber bundle
Let $P\longrightarrow M$ be a $G$ principal bundle endowed with a connection $1$-form $\omega$ (which has values in the Lie algebra $\mathfrak{g}$). If $X_t^x$ denotes Brownian motion on $M$ starting …
- 9,853
7
votes
1
answer
894
views
Sharp Gaussian upper bounds on Heat Kernel
I am looking for references (with proof) for the following statement:
Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compac …
- 5,407
3
votes
1
answer
338
views
Certain construction of the Itô integral on manifolds
Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, …
- 2,703
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
$$\nabla^* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\ …
- 9,853
3
votes
0
answers
406
views
Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly b …
- 9,853
3
votes
The complex heat kernel on a Riemann manifold
As far as I know, the term "Mehler Kernel" is used for the integral kernel of the heat equation corresponding the the harmonic oscillator,
$$ \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 …
- 8,098