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

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91 views

Tetrad transformation

I have been reading an article about Type D gravitational fields, and came across the tetrad transformation that I cannot understand. The author evidently introduced a new coordinate, but just a ...
16
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4answers
493 views

Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?

The title is the question. Sorry, this isn't quite research level. I imagine the answer is well-known, just not to me. Thanks for any help!
1
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2answers
139 views

Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution ...
2
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1answer
115 views

Nowhere vanishing, normalized vector field with bounded derivatives

It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ...
0
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0answers
131 views

Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold. Then Gromov defined the content of the ball by $$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R))) $$ and he remark ...
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0answers
60 views

Largest dimensional Lie subgroup of $SU(N)$ [duplicate]

What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension. I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
8
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1answer
350 views

Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg: Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...
0
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1answer
77 views

About Kahler curvature operator

I have problems on how to consider the Kahler curvature operator. I know that one can consider the Riemannian curvature operator $R$ as a linear transformation from $\mathfrak{so}(n,\mathbb{R})$ to ...
1
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1answer
76 views

Cayley Subspaces in a Calibrated 8-Space

Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces $V$ ...
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0answers
70 views

Proximal point avoids a subvariety

I have a seemingly easy problem about the proximal point in a variety of a general point in the ambient space, which I don't have a proof: Let $X \subset \mathbb{C}^N$ be the affine cone over some ...
1
vote
1answer
127 views

Null geodesic congruence

I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree): Assume we have a Newman Penrose tetrad $\lbrace l, ...
13
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3answers
681 views

Inverse problem of Chern Classes

For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work ...
2
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0answers
114 views

Holomorphic extension of a section of a line bundle

let $(X,g,\omega)$ be a non compact complete K\"ahler manifold of dimension $m\geq3$. Let $\nabla$ the covariant derivative wrt $g$ and $Riem$ the curvature tensor of $g$. Suppose that ...
2
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1answer
120 views

Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function. The classical set up is the following : If $(M,g)$ is a riemannian manifold then we could consider the ...
5
votes
1answer
244 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
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1answer
118 views

Uniqueness of a smooth function [closed]

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
5
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0answers
118 views

Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...
1
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1answer
139 views

Closed parallel (1,1)-forms on compact Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold. We know one example of a closed parallel (1,1)-form, namely, $\omega$ itself. Are there obstructions for the existence of non-vanishing closed parallel ...
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1answer
225 views

symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?
2
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0answers
82 views

Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...
3
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0answers
125 views

On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation $$\triangle \varphi = 4 \pi \rho \qquad (1)$$ defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and ...
2
votes
2answers
160 views

constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...
5
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2answers
321 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
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1answer
126 views

Area of a plane surface that gives a lot of theoretical problems

Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2\subseteq\mathbb{R}^3$ be a injective application, given by: $$\mathbf{r}(u,v)=A(u)+v\cdot (B(u)-A(u)), \forall\ (u,v)\in (a,b)\times (0,1)$$ where ...
2
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0answers
79 views

Sources on evolution of submanifolds subject to Ricci flow

I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome. This old MO post may be ...
2
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0answers
39 views

Hess-Schrader-Uhlenbrock inequality for non-symmetric operators

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some ...
1
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1answer
86 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
5
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1answer
255 views

A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...
3
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1answer
147 views

Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
2
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1answer
141 views

Gromov Geometric Structures and Killing fields

Let's fix some notations: $M$ will denote a real smooth, $m$-dimensional, manifold, $F^k(M)$ is the k-th order frame bundle on $M$ and $Gl^k(m)$ is the space of $k$-jets of diffeomorphisms of $\mathbb ...
5
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0answers
95 views

behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $ ...
0
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0answers
75 views

Contraction with a vector field and pullback bundle

While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ...
5
votes
1answer
196 views

An unusual metric reconstruction problem

$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have ...
13
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1answer
306 views

Is the analytic version of the Whitney Approximation Theorem true?

I initially asked this question on MSE but I haven't had any luck. The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the ...
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0answers
91 views

Canonical metric of toric Kahler manifolds

Let $X$ be non-compact toric Kahler manifold associated to a Delzant polygon $P$ and $g$ be the canonical Kahler metric constructed by Guillemin. Is it true that the real part of $g$, as a ...
6
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2answers
290 views

Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...
16
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2answers
446 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
2
votes
1answer
114 views

Number of connected components of the isometry group of a simply-connected lorentzian manifold

Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie ...
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0answers
163 views

manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...
3
votes
1answer
182 views

Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...
8
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3answers
635 views

is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. We know ...
2
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0answers
133 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
2
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0answers
152 views

Fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by ...
0
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1answer
136 views

Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$. Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...
2
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0answers
74 views

Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates) \begin{equation*} \partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0 \end{equation*} is derived as the gradient flow of the symplectic action functional ...
6
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1answer
275 views

Acyclicity of the sheaf of real analytic differential forms

Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that the Poincare lemma holds for the de Rham ...
12
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1answer
297 views

Harmonic spinors on closed hyperbolic manifolds

Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial? I'm mainly interested in the 3-dimensional case ...
10
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2answers
263 views

Factoring constant rank maps into a submersion and an immersion

Let $X$ and $Z$ be smooth manifolds and $\phi: X \to Z$ a smooth map so that the differential $D \phi$ is everywhere of rank $d$. Is there necessarily a $d$-fold $Y$ so that $\phi$ factors as a ...
5
votes
1answer
200 views

Spectrum of the Laplacian on p-forms on the sphere

In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
2
votes
1answer
160 views

Old Peano theorem (demonstration is missing details) [closed]

Let $A, B :[a,b]\subset\mathbb{R}\to\mathbb{R}^2$ be two functions of class $C^{1}([a,b])$ such that two segments (or intervals) $[A(t_1),B(t_1)]$ and $[A(t_2), B(t_2)]$ never intersect for $t_1, ...