Questions tagged [dg.differential-geometry]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
104 views

A specific question in $G_2$ geometry

Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is \begin{align} d(\theta_{\...
Partha's user avatar
  • 759
1 vote
1 answer
152 views

An identity for the higher form Levi-Civita connection

Take $M$ a Riemannian manifold and $\Lambda^1$ its space of one forms. The LCC (Levi-Civita connection) $\nabla:\Lambda^1 \to \Lambda^1 \otimes \Lambda^1$ is well known to satisfy the identity $m \...
Didier de Montblazon's user avatar
5 votes
2 answers
359 views

Gaussian curvature of a holomorphic curve in complex 2-space

Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$. Each point of $M$ has ...
Daniel Asimov's user avatar
2 votes
0 answers
195 views

When is the Chern integral given by the norm of the curvature tensor?

I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true. $$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$ It ...
Mathgrad's user avatar
  • 293
6 votes
1 answer
346 views

How to differentiate natural transformations?

Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...
Felix Lungu's user avatar
0 votes
0 answers
218 views

Pushforwards in vector bundles over a topological spaces

I have been reading the discussion from Pushforward and pullback.. I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
Nash's user avatar
  • 47
3 votes
2 answers
374 views

Obstructions to the existence of a flat connection on a vector bundle

Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
1 vote
2 answers
151 views

Classification of compact isotropy irreducible homogeneous Kaehler manifolds

Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known? Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible....
Castle's user avatar
  • 21
2 votes
1 answer
252 views

Are there always flat connections?

Let $G$ be a simply connected Lie group and $\Gamma$ a cocompact discrete and torsion-free subgroup. Is there a (real or complex) smooth vector bundle $E$ over the manifold $G/\Gamma$, which does not ...
user avatar
1 vote
1 answer
95 views

Understand Riemannian cross-derivative on product manifolds

Suppose we have a smooth function $f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{R}$ where the domain is a product of two Riemannian manifolds. The Riemannian cross-derivative ([1], section 2) is ...
Jason Li's user avatar
  • 125
0 votes
0 answers
121 views

What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?

I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9): Here, $M$ is a compact Riemannian manifold, $\...
Kaira's user avatar
  • 195
1 vote
0 answers
61 views

Physical measure of a dynamical system in terms of its density

Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ In ergodic theory, the occupation measure is $$\mu_{x, T}(...
NicAG's user avatar
  • 247
12 votes
4 answers
878 views

Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$

I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality: $$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$ where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
stupid_question_bot's user avatar
0 votes
0 answers
50 views

Perpendicular mapping from one matrix group to a closed matrix subgroup

Let H be a connected closed Lie subgroup of G, which is a connected closed Lie subgroup of GL(n,𝔽), where 𝔽 = ℝ or 𝔽 = ℂ. Assume that G and H are each endowed with the riemannian metric inherited ...
Daniel Asimov's user avatar
5 votes
0 answers
357 views

A (possible) Lie algebra extension of the Lie algebra of a foliation

Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
Ali Taghavi's user avatar
4 votes
0 answers
238 views

Dynamical obstruction for a vector field to have a Harmonic divergence

Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
Ali Taghavi's user avatar
1 vote
0 answers
146 views

The space of ergodic elements of a topological or Lie group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
Ali Taghavi's user avatar
2 votes
0 answers
139 views

Understanding the Seiberg-Witten equations in dimension $3$

I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...
Partha's user avatar
  • 759
1 vote
1 answer
256 views

Application of Yamabe and Liouville type equation

Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs): The Yamabe Type Equation (for $n>2$): \begin{equation} -\...
Paul's user avatar
  • 914
3 votes
0 answers
156 views

Properties of the stress energy tensor in Wightman formulation of CFT

In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor: $$\int \mathrm{d}x^...
Connor Mooney's user avatar
5 votes
0 answers
213 views

Algebraic de Rham cohomology with torus coefficients

Let $X$ be a smooth projective variety over $\mathbb{C}.$ On page 3 in this preprint of Simpson, it is stated that Notice first of all that the algebraic de Rham theory is not going to work well in ...
lzww's user avatar
  • 123
5 votes
1 answer
271 views

Lee-Parker Yamabe problem proposition 4.6

I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
Marc's user avatar
  • 312
0 votes
0 answers
49 views

Role of basins of attraction in the Morse decomposition

Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by $$\dot{x}=F(x(t))$$ An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
NicAG's user avatar
  • 247
2 votes
1 answer
117 views

Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?

A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
Shin HY's user avatar
  • 23
1 vote
0 answers
171 views

Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?

Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region? Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
Ali Taghavi's user avatar
3 votes
0 answers
772 views

Is this set a manifold?

Take a general spacetime that is not strongly causal. Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
Bastam Tajik's user avatar
0 votes
0 answers
79 views

Almost Riemannian foliation

A Foliation $\mathcal{F}$ on a Riemannian manifold is called almost Riemannian foliation if $$\forall \epsilon >0 \quad \exists \delta >0 $$ such that for every leaf $L$ and every geodesic $...
Ali Taghavi's user avatar
1 vote
0 answers
90 views

Regularity of minimizing harmonic maps with no topological obstructions

So during (not really) my research I stumbled upon the following question, for which I could not find results in literature in any direction. It is not stated super precisely mathematically speaking, ...
Michele Caselli's user avatar
0 votes
0 answers
38 views

Generic non-existence of 1. Integral of continuous DS

Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by $$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$ In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have ...
NicAG's user avatar
  • 247
2 votes
0 answers
97 views

Deformations of invertible sheaves admitting global sections

We follow Sernesi's treatment of algebraic deformations, working over the complex numbers. Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
Aidan's user avatar
  • 480
2 votes
1 answer
310 views

Bianchi's identity in a principal bundle

Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...
Nabla's user avatar
  • 41
1 vote
0 answers
99 views

Is this a correct description of the BPS monopole of charge $1$?

I am reading the book "The Geometry and Dynamics of Magnetic Monopoles", by M.F. Atiyah and N.J. Hitchin, and I got to this part: "... let $H$ be the Hopf line bundle over $S^2$ and ...
Malkoun's user avatar
  • 4,981
5 votes
1 answer
184 views

Bias of DS literature to polynomial ODEs

In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE $$\...
NicAG's user avatar
  • 247
0 votes
0 answers
98 views

Geodesics in free homotopy classes and the fundamental group

Let $\mathcal{H}$ be the upper half-plane and $\Gamma$ be a cocompact, torsion-free Fuchsian group. The quotient space $X=\Gamma\backslash \mathcal{H}$ is a smooth closed Riemann surface and there is ...
Claudius's user avatar
  • 218
10 votes
1 answer
593 views

Determine whether a (1,2) tensor is Nijenhuis tensor

Given an almost complex structure $J:TM\to TM$, the Nijenhuis tensor $N_J:\wedge^2TM\to TM$ is given by $N_{J}(X,Y)=[X,Y]+J([JX,Y]+[X,JY])-[JX,JY]$. My question is, is there a necessary and sufficient ...
Liding Yao's user avatar
0 votes
0 answers
88 views

Affine manifold and topology

I consider an affine manifold $(M,\nabla)$, i.e. $\nabla$ is flat and torsion-free, such that: $M$ is diffeomorphic to $\mathbb R\times\Sigma$ with $\Sigma$ a closed 3-manifold. There exists a ...
Vigneron Quentin's user avatar
3 votes
1 answer
379 views

Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?

Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
Paul Cusson's user avatar
  • 1,735
5 votes
2 answers
229 views

Does a $C^1$ perturbation induces diffeomorphic level set?

Consider a proper $C^1$-function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ where $n\geq m$. If $c\in \mathbb{R}^m$ is a regular value of $f$, then we know that $f^{-1}(c)$ becomes a compact submanifold ...
GHG's user avatar
  • 173
2 votes
0 answers
116 views

Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
53Demonslayer's user avatar
4 votes
2 answers
325 views

Holomorphic Gauss normal map

Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$. Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic ...
Ali Taghavi's user avatar
3 votes
1 answer
198 views

Convergence of spectrum

Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$. Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
Hammerhead's user avatar
  • 1,171
1 vote
0 answers
116 views

Non-compact extremal Kähler spaces

I want to ask about a generalisation of the Calabi functional to non-compact Kähler spaces. My interest is mostly in Kähler surfaces, so I will assume real dimension $4$. In my work, I have found an ...
Sergei Ovchinnikov's user avatar
0 votes
0 answers
98 views

Under what condition can a smooth map be factored through the Gauss normal map

Inspired by this question entitled When does the shape operator commute with a derivative? we ask the following question: Assume that $S,H$ are two surfaces whose corresponding Gauss maps are denoted ...
Ali Taghavi's user avatar
0 votes
0 answers
295 views

Proof that a first integral is not a constant function

Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions $$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$ such that all of them are differentiable and ...
NicAG's user avatar
  • 247
2 votes
1 answer
99 views

Why is this subset associated to a $2$-tensor dense?

Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
Matheus Andrade's user avatar
0 votes
1 answer
483 views

Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
Bastam Tajik's user avatar
0 votes
0 answers
100 views

Lorentzian geometry. Comparing Honda's main theorem A construction to mine: Mixed type surfaces

This question is based on a wonderful paper by A. Honda (link below) where his main theorem A provides an incredible uniqueness result. Mixed type surfaces and type changing metrics have been ...
53Demonslayer's user avatar
2 votes
0 answers
110 views

The inverse of the metric on the space of harmonic p-forms

Let $X$ be a manifold equipped with a riemannian metric $g$. The space of harmonic p-forms is an $b_{p}(X)$ dimensional vector space with $b_{p}(X)$, the pth Betti number. One can then define a metric ...
dbrane's user avatar
  • 141
3 votes
0 answers
102 views

Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields

Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection. ...
Ali Taghavi's user avatar
22 votes
1 answer
2k views

A difficult integral for the Chern number

Cross post from Maths stack exchange The integral $$ I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
xzd209's user avatar
  • 323

1
3 4
5
6 7
173