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Questions tagged [dg.differential-geometry]

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

13 questions from the last 7 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....
Jonathan's user avatar
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0 answers
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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 "...
Matteo Raffaelli's user avatar
7 votes
1 answer
301 views

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 ...
Matthew Kvalheim's user avatar
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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 ...
Learning math's user avatar
-1 votes
1 answer
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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.
Vertvolt's user avatar
1 vote
0 answers
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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 ...
Tyrannosaurus's user avatar
5 votes
0 answers
93 views

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 ...
Jan Heck's user avatar
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87 views
+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}...
Student's user avatar
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8 votes
0 answers
238 views
+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 ...
Matthew Kvalheim's user avatar
3 votes
0 answers
101 views

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 ...
mlainz's user avatar
  • 161
3 votes
0 answers
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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,...
Seba's user avatar
  • 126
4 votes
0 answers
71 views

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 ...
Phil-W's user avatar
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1 vote
0 answers
39 views

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 $$ '\...
neander's user avatar
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