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

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14
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1answer
166 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 ...
0
votes
0answers
49 views

Why $C^0$ - Convergence gives Gromov-Hausdorff convergence?

Let $(M,\omega (t)) $ be a family of kahler forms , why $C^0$ - Convergence gives Gromov-Hausdorff convergence?
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 ...
-1
votes
0answers
37 views

Problems concerning meromorphic 1 form on Riemann surface [on hold]

1. For a compact Riemann surface $X$, let the genus of $X$ be the dimension of the space of holomorphic one forms on $X$. How to compute the genus of the Riemann sphere and a complex torus. 2. ...
0
votes
0answers
48 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?
0
votes
0answers
66 views

Does a (non-closed) differential 1-form define a curve? [on hold]

I am trying to understand under what conditions the following procedure properly defines a curve. Take a manifold $M$ with a (non-closed) 1-form $B$ and an exact 1-form $dA$. Define the functions ...
2
votes
0answers
58 views

Lower boundedness of the Ricci curvature [on hold]

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 ...
0
votes
2answers
138 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 ...
4
votes
2answers
453 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 ...
-5
votes
1answer
135 views

Stiefel-Whitney class of complex projective spaces [on hold]

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
151 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
0answers
42 views

Two surfaces with zero gaussian curvature [on hold]

According to Hartman's article every surface $f(s,t)$ with zero gaussian curvature locally admits parametrization $f = a_1(u) v + b_1(u),$ $s = a_2(u) v + b_2(u),$ $t = a_3(u) v + b_3(u).$ Now let's ...
0
votes
1answer
47 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
211 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
67 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
137 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
68 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
181 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
125 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
1answer
89 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
150 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 ...
3
votes
1answer
83 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
165 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 ...
13
votes
2answers
604 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
69 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
186 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
55 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
118 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
65 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
131 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
225 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
251 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
504 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 ...
2
votes
0answers
133 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
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0answers
68 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
126 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
132 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 ...
4
votes
0answers
150 views

Curvature estimates for Kaehler-Einstein and Hermitian, Einstein four-manifolds

Four-dimensional Kaehler-Einstein manifolds and Hermitian, Einstein manifolds with positive scalar curvature have been classified by Tian and LeBrun respectively. I was wondering are there any ...
1
vote
1answer
85 views

Uniqueness affine curvature

Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...
3
votes
0answers
179 views

Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it? I guess maybe there are two competing settings a priori: a complex ...
0
votes
0answers
59 views

Bispinors, polyforms, bilinears and supersymmetric manifolds

Let $(V,q)$ be a regular quadratic vector space, and let us denote by $Cl(V,q)$ the corresponding Clifford algebra. Then there exists an isomorphism of $\mathbb{Z}_{2}$-graded algebras: ...
2
votes
1answer
136 views

Closed geodesics that cross one another frequently

Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$. $S$ has at least three simple (non-self-intersecting), closed geodesics by a theorem of Lyusternik and Shnirel'man. Alternatively, ...
1
vote
1answer
117 views

Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles

As explained in: Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their ...
1
vote
0answers
97 views

Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula ...
1
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0answers
138 views

Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?
3
votes
0answers
48 views

Geodesic rays in a toric variety

Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...
4
votes
3answers
250 views

Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
1
vote
0answers
104 views

Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...
1
vote
1answer
120 views

Is there a name for the function on $TTM$ swapping the 2nd and 3rd coordinates?

I'm not so good on geometry, so I fear this is a relatively basic question. For any $N \in \mathbb{N}$, let us identify the tangent bundle of $\mathbb{R}^N$ with $\mathbb{R}^{2N}$ in the obvious ...
0
votes
0answers
51 views

Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...