Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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4
votes
2answers
355 views

How many metrics of constant curvature exist on a Riemannian surface?

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...
2
votes
0answers
102 views

Heat semigroup estimate on complete Riemannian manifold

Consider a complete noncompact Riemannian manifold $M$ such that the heat kernel $h_t(x, y)$ satisfies $h_t(x, y) \leq Ct^{-n/2}$. Consider a function $u \in L^p(M)$. How can we prove that ...
7
votes
3answers
284 views

Classification of natural invariants of Riemannian structures

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi. By ...
3
votes
1answer
118 views

convex decompositions of the sphere

Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...
1
vote
1answer
135 views

Metric, torsion free connections on principal bundles

I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers. Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold ...
4
votes
0answers
96 views

Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence

Thank you for spending time on the following question. I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic ...
2
votes
1answer
173 views

Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected. Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...
0
votes
0answers
85 views

When does the integral of a Dolbeault-exact form vanish?

What conditions (if any) can be imposed on a Kahler manifold $M$ so that we get a Dolbeault analogue of Stokes' theorem on a closed manifold, i.e. $\int_M \partial ( ... ) =0$ The trivial solution ...
2
votes
1answer
108 views

Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact). Question 1. Is it true that every point of $X$ has a ...
1
vote
0answers
54 views

Generalize Gauss-Bonnet Formula to non-simple closed curves [migrated]

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense: For a domain $\Omega$ enclosed by an non-simple closed curve ...
3
votes
1answer
171 views

A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any ...
1
vote
0answers
39 views

Invariant Girsanov Theorem on a Riemannian manifold

This is somewhat a follow-up on this post. Let $X_t$ be the stochastic process on a compact Riemannian manifold generated by the (possible time-dependent) second-order elliptic operator $L_t$. Let ...
1
vote
1answer
224 views

Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$. We have the ...
3
votes
1answer
84 views

Null tetrad transformation

I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism. I have a question about what he calls a "class III ...
6
votes
2answers
326 views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...
13
votes
2answers
647 views

(Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
2
votes
1answer
130 views

Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered. Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...
24
votes
2answers
721 views

Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...
1
vote
0answers
49 views

freedom in choosing a smooth function of compact support [migrated]

Suppose $\Omega$ represents a bounded domain in $\mathbb{R}^n$. For ease, let $\Omega$ be a ball around the origin. Let $\varphi$ be a smooth radial function defined on $\Omega$. I am trying to ...
0
votes
0answers
53 views

Derivative of Log_p map

If $\mathscr{M}$ is a $(d-dimensional)$ Riemann Manifold and $p$ is a point therein. What is the derivative of $Log_p$ function?
2
votes
0answers
73 views

Lower boundedness of the Ricci curvature [closed]

I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The ...
1
vote
3answers
252 views

orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$. I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates. For ...
7
votes
2answers
516 views

Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...
-6
votes
1answer
170 views

Stiefel-Whitney class of complex projective spaces [closed]

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
1
vote
1answer
165 views

Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...
0
votes
1answer
53 views

Principal bundle associated to a Courant algebroid

A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition ...
4
votes
1answer
233 views

Differential forms along the fiber

Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...
1
vote
1answer
89 views

Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...
0
votes
1answer
159 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
4
votes
0answers
73 views

Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
5
votes
1answer
202 views

Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ ...
3
votes
2answers
193 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
0
votes
0answers
128 views

Existence of a special Kahler structure on the contangent bundle

It is known that the total space of the cotangent bundle $T^{\ast}M\to M$ of any manifold $M$ can be equipped with a Kahler structure. My question is, it is known when $T^{\ast}M$ admits a Special ...
2
votes
2answers
163 views

Connections having the same holonomy along loops at a point

I find myself stuck with the following question, which seems very classical but for which I have not been able to find a reference. Consider a smooth vector bundle $E$ of rank $r$ over a compact ...
4
votes
1answer
147 views

Lagrangian submanifold of a Calabi-Yau manifold

In the paper 'Special Lagrangians, stable bundles and mean curvature flow' by R. P. Thomas and S.-T. Yau, page 2. They said A Lagrangian submanifold $L$ of the Calabi-Yau manifold $(X,\Omega)$, we ...
3
votes
3answers
184 views

Are linearizations of involutive PDEs locally solvable?

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing ...
14
votes
2answers
646 views

Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...
2
votes
1answer
79 views

distance formula of warped products

Given a warped product, I want to compute the ditance of any two points. First get the equation for the geodesic, then compute the length of the geodesic. Consider the two-dimensional surface $$ ...
1
vote
1answer
200 views

Is there a way to define a Lie derivative of a connection?

I've been reading a little bit about the definition of symmetries on General Relativity, and they are related with the concept of Killing vector, i.e., vectors along which the Lie derivative of the ...
2
votes
0answers
65 views

Analogue of the Euler class of a circle bundle and the global angular form

This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle. For an SO(2) ...
3
votes
1answer
128 views

structure of metrics on a compact manifold

is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $? i have a given manifold $M$, a given measure $\mu$ with an everywhere positive ...
0
votes
0answers
75 views

Functional involving Ricci curvature: convex and coercive?

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, volume form $\mu_g$, and Ricci curvature $\text{Rc}_g$. Question: Given a fixed vector field $V\in\Gamma(TM)$, under what ...
1
vote
1answer
139 views

Symplectic reduction: from indefinite signature to Riemannian signature

Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...
3
votes
1answer
232 views

'Unitary' charts on odd-dimensional spheres

Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific ...
2
votes
1answer
261 views

Perimeter of ellipse: Combination of two geometries

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter ...
10
votes
1answer
521 views

Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula $$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$ He gives ...
3
votes
0answers
143 views

Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data: $$(1) \ \ ...
0
votes
0answers
70 views

Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...
1
vote
0answers
139 views

How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions. ...
2
votes
1answer
140 views

Is there a characterization of Riemannian manifolds that split off two factors?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...