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Questions tagged [dg.differential-geometry]

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

1
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3answers
143 views

Fibered product of stacks comes from a Lie groupoid

I am adding some context here. I am reading Introduction to Differentiable Stacks by Gregory Ginot. In page no $7$, just before the remark $2.2$ he says the following. One shall be careful that ...
4
votes
1answer
54 views

Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...
3
votes
3answers
273 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
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0answers
39 views

Is it okay to consider two courses in differential geometry the same semester [on hold]

Is it a good idea to take introduction to differential geometry (using De Carmo book) and Calculus on Manifolds (using Lee Jeffery's book) in the same semester?
4
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1answer
139 views

Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $ \theta:G\rightarrow H$  and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle. See that the morphism of Lie ...
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0answers
83 views

Is conjugation by gauge transformation of $G$-bundle contained in $\mathfrak{g}$?

Before painstakingly defining all these terms, let me ask my question in plain english: given a $G$-bundle, is every conjugate of a vector field by a gauge transformation an element of the Lie algebra?...
9
votes
1answer
208 views

k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$. Question. Are ...
19
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1answer
290 views

Is the metric completion of a Riemannian manifold always a geodesic space?

A length space is a metric space $X$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $c: [0,1] \rightarrow X$ is the sup of $$ d(c(0),...
19
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3answers
528 views

Integral cohomology of $SU(n)$ - looking for constants

I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$,...
4
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0answers
130 views

What is variation of the Chern-Simons functional, and why can it be calculated as follows?

Let $G$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $X,$ we let $\...
1
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2answers
168 views

Almost complex structure and intrinsic torsion

Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$. I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$...
2
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1answer
248 views

Is there a theorem showing that de Rham homology is isomorphic to singular homology?

The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology. Now ...
3
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0answers
122 views

Examples of incomplete Lorentz 3-manifolds

Reading this paper where closed 3-dim. Lorentz manifolds with noncompact isometry groups are studied, I wonder if all of them are geodesically complete. One class of 3-dim. closed Lorentz manifolds ...
2
votes
1answer
97 views

Local Sobolev embedding on complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball. Q Can we find a constant $C=C(\kappa,r,m)$(...
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0answers
82 views

A geometrical problem in terms of a convex function

I wish to know whether the following problem has ever been investigated. Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
10
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0answers
137 views

Analogy between BV formalism and integration by residues

Domenico Fiorenza begins his description of the Batalin–Vilkovisky formalism by pointing out an analogy with integration by residues: Take a top form (density) on $\mathbf R$ resp. space of fields; ...
21
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15answers
3k views

Geodesics on the sphere

In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ...
5
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1answer
224 views

Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average{...
2
votes
1answer
182 views

Understanding the definition of $G$-gerbe

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following. Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
13
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3answers
521 views

Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set $$ \operatorname{perim}(S)=\| \nabla 1_S\|_{TV} $$ ...
31
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1answer
895 views

Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...
2
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0answers
83 views

Central extension gives a gerbe over stack

Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$. I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism ...
2
votes
1answer
86 views

regularity of harmonic forms on manifolds-with-boundaries

Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
3
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1answer
144 views

Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma

I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
1
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1answer
96 views

Poisson equation on noncompact manifold

Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space. For the equation $$\Delta u=f,$$ ...
0
votes
1answer
82 views

Analytic approach to geodesic connectedness in Semi-Riemannian manifolds

Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
0
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0answers
57 views

Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...
2
votes
1answer
73 views

Increasing union of embedded submanifold is immersed manifold

While working on the proof of the stable manifold theorem, I came across a problem that I'm not able to really grasp. Given some Anosov map $f: M \to M$ on a compact Riemann manifold $M$, one can ...
13
votes
1answer
290 views

Projective-invariant differential operator

This question was originally asked on Math StackExchange. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = ...
1
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0answers
58 views

$C^1$-foliation are absolutely continuous

Brin & Stuck defined in Introduction to dynamical system two notions: That of a absolutely continuous foliation : given any foliated chart $U$ on some Riemannian manifold $M$ (with foliation $W$),...
3
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0answers
105 views

About Minkowski's problem

Let $f$ be a positive function over the unit sphere $S^{d-1}$. Minkowski's problem is to find a convex body $K$ in ${\mathbb R}^d$, whose Gauss curvature is prescribed as a function of the normal ...
0
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0answers
39 views

Dirichlet problem and schauder estimate for manifolds

Let $M$ be an n dimensional Riemannian manifold without boundary. Let $U\subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(U)$, $g\in C^{2,\alpha}(\partial U)$. Consider the ...
10
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1answer
386 views

Serre spectral sequence for de Rham cohomology

Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with $$ E_2^{p,q} = H^p(M,\underline{H^...
6
votes
2answers
457 views

Lower regularity version of Moser's theorem on volume elements

A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; doi: 10.1090/S0002-9947-1965-0182927-5, jstor), shows that if a $C^...
6
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0answers
108 views

Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $$ ...
6
votes
1answer
108 views

Normal coordinates for isotropic submanifolds

Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ in a neighbourhood $U$ of $p$ such ...
6
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0answers
227 views

Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
9
votes
2answers
1k views

Asymptotic comparison of $L^2$ sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$. For $m$ large, let $\{e_1,\...
5
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0answers
65 views

Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
0
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0answers
84 views

Intrinsic Reach for a Riemannian manifold

The reach of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$. My question: ...
0
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0answers
39 views

What is the meaning and role of global hyperbolicity condition for semi-Riemannian manifolds

What is the heuristic meaning of the global hyperbolicity condition for semi-Riemannian manifolds? Also what is the role of this condition in the study of geodesic connectedness?
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3answers
338 views

Are rotations generated by translations, scalar multiplications and inversions? [closed]

I read a paper saying the Mobius transformation from $\mathbb{R}^n\cup \infty \to \mathbb{R}^n\cup \infty$is generated by translations, scalar multiplications and inversions $x\to \frac{x}{|x|^2}$. So ...
2
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0answers
141 views

Examples of certain compact Kaehler manifolds

A Kaehler manifold is a complex manifold which has a Kaehler metric and Ricci curvature tensor $R_{ij}$. The Ricci curvature tensor is a Hermitian matrix having real eigenvalues. My question is: Is ...
8
votes
1answer
190 views

Isometry group of low dimensional Alexandrov space

It is known by the work of Galaz-García and Guijarro, that the dimension of the isometry group of an $n$-dimensional Alexandrov space (of curvature bounded below) is bounded above by $\frac{n(n+1)}{2}$...
2
votes
0answers
92 views

(Singular) metric associated to the higher cohomology

Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$. ...
0
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0answers
83 views

(Semi-)Riemannian geometry for working PDE analysts

What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)? The closest thing I know to this, are two books by ...
1
vote
2answers
413 views

Reference for some elementary facts about principal bundles

Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$. Now I am searching for a good reference (with proofs) for the following facts: 1) ...
0
votes
0answers
34 views

Critical metric for an Hilbert action?

Suppose that $\omega$ is an 1-form on a Reimannain Manifold $(M,g)$ and $s$ is a $(0,2)$ symmetric tensor which be considered as $(1,1)$ symmetric tensor whenever it is convenient. If for all $s$ the ...
6
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0answers
165 views

Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...
5
votes
1answer
212 views

Conversion formula between “generalized” Stiefel-Whitney class of real vector bundles: O(n) and SO(n)

$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, $$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$ Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as: $$ w_j(...