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

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0
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32 views

Generalized Plateau problem with non-Jordan boundary

Let $C_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A_\pm:(R,0,\pm 1)$ and $B_\pm:(-R,0,\pm 1)$. Assume that ...
0
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0answers
14 views

Measures for the Eccentricity of General Strictly Convex Smooth Closed Manifolds of Genus 0

Question: Are there any measures for how much the shape of a strictly convex smooth closed manifold of genus 0 deviates from that of a hyper-sphere of equal dimension? In euclidean 2-space ...
1
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0answers
79 views

Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following: Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...
1
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0answers
16 views

Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...
13
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1answer
521 views

What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?
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0answers
45 views

How do the heat kernel coefficients depend on the curvature tensor

this is a crosspost, the same question was asked first here: http://math.stackexchange.com/questions/1640092/polynomial-in-the-components-of-the-curvature-tensor Since I have not received any answers ...
-2
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0answers
24 views

Loxodrome loop on a surface of revolution [on hold]

Prove that there can be a closed loxodrommic curve on a surface of revolution only when it is doubly connected.
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0answers
61 views

Intuitive understanding of the mean curvature flow [on hold]

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = ...
10
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0answers
93 views

Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces $$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction. Richard ...
6
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1answer
155 views

Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
8
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2answers
238 views

Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space: Therefore the quotient manifold $$ \mathbb{HP}^{2}/\mathrm{U}(1) $$ may be taken, writing $U(1)$ for the circle group. It has ...
1
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2answers
114 views

Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
2
votes
1answer
110 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
3
votes
1answer
169 views

Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
2
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0answers
68 views

Is it possible to write down the explicit expressions of some extensions of conformal vector fields on spheres?

Let X be a conformal vector field on the standard sphere $S^n$ with standard metric $g_{S^n}$, then there exists a unique conformal vector fields in the unit ball $B_1(0)\subset \mathbb{R}^{n+1}$, ...
2
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1answer
86 views

Lifting a differential operator

Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
6
votes
1answer
275 views

Solutions of equations characterizing a complex structure

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ ...
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0answers
93 views

Differential geometry [closed]

If we have integrable distribution D of rank k on a manifold, and we have k functions which are zero homogeneous and constant on the leaves (basic functions). Can we glue together these functions to ...
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0answers
51 views

Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
3
votes
1answer
207 views

Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...
4
votes
1answer
180 views

The fibration map $Diff(M) \rightarrow Emb(N,M)$

Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map $$ Diff(M) \rightarrow ...
4
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0answers
98 views

Moduli space of complex Tori [duplicate]

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?
4
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0answers
72 views

Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows: Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
28
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3answers
2k views

What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...
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570 views
+200

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
5
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2answers
147 views

No normal coordinates on general Finsler manifolds

I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates. As I'm still new to non-Riemmanian Finsler geometry I don't see why ...
1
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1answer
77 views

Dirac bundle and spinor bundle

What is the difference between Dirac bundle and spinor bundle?Morever, every spinor bundle is Dirac bundle, is it true?
5
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1answer
553 views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
5
votes
2answers
300 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
2
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0answers
79 views

Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [closed]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
4
votes
1answer
97 views

Glueing together functions defined on the leaves of a foliation

Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying. Consider a Poisson ...
1
vote
1answer
69 views

Nash-type theorems for Poisson manifolds

My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am ...
5
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0answers
51 views

How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
22
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1answer
227 views

Height function on 2-torus with only 3 critical points

It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...
5
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0answers
115 views

The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold

Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on ...
5
votes
1answer
97 views

Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
4
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0answers
142 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
6
votes
1answer
386 views

Is there a Nash-type theorem for symplectic manifolds?

If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)? ...
0
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0answers
59 views

Weak $H^1$-limit of “almost conformal” maps

Let us consider a sequence of maps $\phi_n : (M, g) \to \mathbb{R}^k$, where $M$ is a surface. Let $\phi_n \to \phi$ weakly in the sense of $H^1$-norm, and let $\phi$ be non-trivial. Let $\mathcal{H}$ ...
4
votes
2answers
182 views

Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi ...
3
votes
1answer
78 views

Parallel Transport on Hypersurface Spinor Bundle

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link: ...
2
votes
2answers
179 views

Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function ...
1
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1answer
51 views

Marginally Trapped surfaces with constant Gaussian curvature

By marginally trapped surface I mean a spacelike surface in a 4-dimensional Lorentzian manifold such that the mean curvature vector is lightlike. In my research I have stumbled across marginally ...
7
votes
1answer
294 views

Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$. The Cartan connection is supposed to formalize what it means to "roll ...
0
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0answers
69 views

If the fibers of a submersion are connected, does it mean that any 2 sections are homotopic (locally on the base)?

Is the following fact known? If yes - what is the reference? Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers. Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any ...
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0answers
56 views

Do the germs of local vector fields which are generated by a Lie pseudogroup form a Lie algebra?

Let $(M,g)$ a semi-Riemannian manifold, and let $\mathcal{G}$ a be Lie pseudogroup which acts by local isometries on $(M,g)$. Clearly the germs of all local killing vector fields at $p \in M$ form a ...
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0answers
63 views

Euclidean metric in spherical coordinates [closed]

I apologize if this question is not research level, but it has been asked on MSE before, and not received really satisfactory answers, for example, see here, and googling does not reveal anything ...
4
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0answers
420 views

geodesic computation: “energy” minimization versus arc length minimization [migrated]

Is it true that applying the Euler-Lagrange equation to the integral $E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length ...
6
votes
2answers
241 views

Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
3
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0answers
92 views

Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity. Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...