Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
36 questions from the last 30 days
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Is there a symbolic computation program that can deal with differential forms, stokes theorems, the hodge * operator, etc
I am looking at a messy series of computations of integrals of differential forms on manifolds with boundary, involving repeated application of Stokes' theorem, and also involving the Hodge * operator....
- 1
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22
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 965
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1
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298
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Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 491
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32
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,465
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1
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On the correspondence between infinitesimal and integral description of connections
It is the title of an article by Petko Nikolov Triste Sissa 1981. I cannot access this pdf yet I remember that it was once avaliable on libgen and now I cannot find it. Please help.
- 21
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44
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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 168
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93
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Is the pullback of differential forms on a compact manifold smooth tame as a map of Fréchet manifolds?
In Hamilton's paper on the Nash-Moser inverse function theorem he shows that if $M$ is a smooth compact manifold and $V\to M$ a smooth vector bundle then its smooth sections $\Gamma(V)$ equipped with ...
- 51
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85
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+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 537
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228
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+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 491
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100
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Understanding the Lie derivative by multivector fields
For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one ...
- 161
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Orbit space of the action of $\mathrm{GL}(V)$ on the Grassmannian of $V\wedge V$
$
\newcommand{\K}{\mathbb{K}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Grass}{Grass}
$Consider $\K\in\{\R,...
- 126
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69
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Integration of volume forms over manifolds with corners
Suppose that $M$ is a (compact, oriented, smooth) manifold with corners.
Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
- 1,035
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39
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
- 161
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1
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Almost Complex Structure extending to Complex Structure, aka "Integrable"
Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \...
- 6,018
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Classification of principal $\mathrm{SO}(3)$-bundles on a 4-manifold via characteristic classes
I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:
Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are ...
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2
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300
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+500
Cohomology of foliations and closed forms along the leaves
Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
- 2,816
3
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1
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269
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$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
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66
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Interplay Between Curvature, Volume, and Optimization: Extending Results Beyond Surfaces of Revolution in $\mathbf R^3$ [closed]
Recently, I discovered a precise formulation directly relevant to my research:
Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique codimension one surface of revolution, $L$, with a complete ...
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Wedge of curvature and subsequent trace
I am currently reading https://arxiv.org/abs/1901.10322. More specifically, I am interested in understanding the equation
$$i\partial\overline{\partial}\omega = \frac{\alpha'}{4}Tr(R\wedge R-F\wedge F)...
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2
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332
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Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
I am looking for proof of the "well-known" result that for a $4$-dimensional Riemannian manifold $(M, g)$, we have an isomorphism
$$
\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
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On the trajectory followed by a point P on a planar convex region C when P is mapped repeatedly to the farthest point to it on C
Consider a planar convex region $C$. Let us define a mapping of a point $P$ on $C$ to that point on C that is farthest from $P$. Obviously, if from an initial position of $P$, we do this mapping ...
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178
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Recognize this metric? Do you have a name for this metric on the product of spheres?
Take the product $S^2 \times S^2$ of two two-spheres,
but perturb the product metric as follows.
Think of each $S^2$ as the unit two-sphere in Euclidean 3-space
in the standard way
so that for $p ...
- 8,336
2
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0
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106
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The existence of a harmonic diffeomorphism on a punctured surface
Let $\bar{X}$ be a compact Riemann surface of genus $g$, and let $D:=\{p_{1},p_{2},\cdots,p_{n}\}$ be $n$ distinct points on $\bar{X}$. Define $X:=\bar{X}-D$ to be the punctured Riemann surface given ...
- 391
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1
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172
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Mass minimizing current in real homology class
It is a well-known results by Federer and Fleming that there exists at least one mass-minimizing normal current in every real homology class of a closed $n$-dimensional Riemannian manifold $M$. Their ...
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1
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547
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Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
2
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1
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Causality of Killing vector fields in a Lorentzian Ricci flat spacetime
In a connected Lorentzian spacetime that is Ricci flat and also nice (say smooth, global hyperbolic, etc), can a global Killing vector field be null in an open subset and timelike (or spacelike) in ...
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Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
- 21
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1
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213
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Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
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179
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Deformations of cotangent bundles
Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, ...
- 6,622
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323
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An inequality that may be of isoperimetric nature
I am trying to prove the following inequality: let $f,g:S^1\to R$ (here $S^1$ is the unit circle parametrized by arc-length) be differentiable and have zero mean. Then
$$
4\pi \int f(t) g(t)\, dt \le \...
- 797
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1
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158
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Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
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Is every connection locally flat for an other connection?
Consider a $C^{\infty}$ connection $d_A = d+A$ on the unit ball $B^n\subset \mathbb{R}^n$. Does there exists another connection $d_{\tilde{A}} = d+\tilde{A}$ such that $d_{\tilde{A}} A = 0$? That is ...
- 363
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1
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380
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Is it always possible to connect the endpoints of a smooth injective path, so the resulting path is smooth, closed and simple?
Motivation
The motivation for this question arises from the following problem: Is there a closed, simple, and infinitely differentiable path on the (complex) plane such that every straight line has a ...
5
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2
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299
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Non-semisimple Lie groups and Higgs bundles
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $X$ be a compact Riemann surface. Let $G$ be a real reductive Lie group, $H$ be a maximal compact subgroup of $G$ ...
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2
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424
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Questions about some parallel between polynomial and differential equation
Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations ?
Do the relations between ...
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65
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Axes of symmetry and symmetry group of the tangent cone to an open, connected, convex subset of the Euclidean space
Given a closed convex set $K\subset \mathbb{R}^d$ and a point $x\in K$ the tangent cone to $K$ at $x$ is defined by
\begin{equation}
T_xK:=\overline{\{v\in \mathbb{R}^d: \exists \lambda \geq 0 \text{ ...
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