# Tagged Questions

**3**

votes

**1**answer

138 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 ...

**1**

vote

**0**answers

29 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 ...

**2**

votes

**0**answers

109 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**

vote

**0**answers

22 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 ...

**15**

votes

**1**answer

816 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?

**1**

vote

**0**answers

84 views

### Intuitive understanding of the mean curvature flow [closed]

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 = ...

**16**

votes

**1**answer

302 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 ...

**7**

votes

**1**answer

193 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**

votes

**2**answers

270 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**

vote

**2**answers

129 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

**1**answer

160 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 ...

**4**

votes

**1**answer

191 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**

votes

**0**answers

85 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 $B_1(0)\subset \mathbb{R}^{n+1}$, ...

**3**

votes

**1**answer

100 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 ...

**9**

votes

**1**answer

795 views

### Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of ...

**0**

votes

**0**answers

32 views

### Approximation of a volume-preserving Hölder homeomorphism by diffeomorphisms?

Is it known whether a volume-preserving Hölder homeomorphism of an arbitrary manifold can be approximated by a volume-preserving diffeomorphism?
The answer is clearly no if the volume-preserving ...

**0**

votes

**0**answers

55 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

**1**answer

246 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

**1**answer

198 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**

votes

**0**answers

104 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?

**5**

votes

**0**answers

96 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 $S$ ...

**30**

votes

**3**answers

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 ...

**17**

votes

**0**answers

767 views

### 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**

votes

**2**answers

184 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**

vote

**1**answer

94 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?

**8**

votes

**1**answer

836 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

**2**answers

330 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**

votes

**0**answers

85 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

**1**answer

107 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

**1**answer

77 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 ...

**6**

votes

**0**answers

96 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 $z=x+ i y \in \mathbb C$ ...

**23**

votes

**1**answer

284 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**

votes

**0**answers

125 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

**1**answer

114 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**

votes

**0**answers

154 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

**1**answer

406 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**

votes

**0**answers

62 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

**2**answers

190 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

**1**answer

89 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

**2**answers

199 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**

vote

**1**answer

60 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 ...

**8**

votes

**1**answer

319 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**

votes

**0**answers

143 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 ...

**0**

votes

**0**answers

58 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 ...

**1**

vote

**0**answers

67 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 ...

**6**

votes

**2**answers

251 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**

votes

**0**answers

101 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 ...

**7**

votes

**2**answers

224 views

### What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?

Physicists routinely wrote all 3 Pauli spin matrices as a vector.
$$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma_2 = \left( ...

**2**

votes

**2**answers

157 views

### Intersection of Subspaces with $O(3)$

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.
For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...

**4**

votes

**0**answers

100 views

### Deligne-Hitchin twistor space

Let $M$ be the moduli space of stable holomorphic Higgs bundles on a Riemann surface $\Sigma$. By Hitchin s work it is equipped with a hyper-kaehler structure, and one can define its twistor space, a ...