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

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2
votes
1answer
93 views

Modification of Morse lemma with two functions

Standard Morse lemma states that if the singularity of a smooth function $f$ is non-degenerate, one can choose coordinates such that function has a "quadratic" form $f = \sum \limits_i x_i^2 - \sum ...
1
vote
1answer
121 views

Is a non-flat hermitian connection determined uniquely by its holonomy and curvature?

How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where $$\alpha\in\text{Hom}(\pi_1(M),U(1)),~~~~F\in ...
1
vote
1answer
166 views

Tangent complex of dgla/ twisted dgla

I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism. Let $(\mathfrak{g}, d, ...
2
votes
0answers
117 views

Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$? Think of $\Bbb R ...
2
votes
1answer
135 views

Can one smooth open star shaped domains from the inside by star shaped domains?

Let $O\subset\mathbb{R}^n$ be a open set which is star shaped with respect to the origin. How does one prove that there exists an increasing sequence of star shaped (w.r.t the origin) domains $O_i$ ...
7
votes
0answers
115 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 ...
1
vote
0answers
44 views

Convex Hull and Least Area Discs in Riemannian 3-Manifolds

Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all discs bounded by $\gamma$. Let ...
3
votes
1answer
165 views

How many second-order PDEs can be obtained from a contact EDS?

Let $(M,[\theta])$ be a contact manifold, $\dim M=2n+1$, and denote by $\mathcal{I}^\theta$ the differential ideal generated by the contact form $\theta$. An exterior differential system on $M$ of ...
1
vote
1answer
251 views

Ordinal of injectivity for a smooth regular curve with a finite arc-length

Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve ...
5
votes
1answer
138 views

Affinely flat structures. How many different ones on the same manifold?

Let $M$ be a smooth manifold endowed with $\nabla$ a flat torsion-free connection. Let $\operatorname{Diff}(M)$ be the group of smooth diffeomorphisms of $M.$ Obviously if $ \phi \in ...
1
vote
0answers
49 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$ [closed]

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...
3
votes
0answers
57 views

Modify the jump set of $BV$ function

Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
2
votes
1answer
110 views

Regularity - mean curvature equation

In my research I arrived at the following equation: $$ \int_B \frac{\nabla u \cdot \nabla \varphi}{ \sqrt{1+|\nabla u|^2}}=\int_B f \varphi, (*)$$ for every $\varphi \in C^1(B)$, which is a weak form ...
2
votes
0answers
154 views

Ricci curvature in resolution of singularities

Let $X$ and $X'$ are Kahler variety and $f: (X',\omega')\to (X,\omega)$ be the resolution of singularities of $X$ then from $K_X=f^*K_X'+E$ how can we find the relation between $Ric(\omega)$ and ...
2
votes
0answers
407 views

On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) ‎‎\subseteq‎‎ TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. ‎L‎‎et $\bar{J} : TTM \longrightarrow ...
4
votes
1answer
142 views

Elliptic Operators on Vector Bundles

I know the kernel of an elliptic operator on a compact manifold has finite dimension. Is the kernel of an elliptic operator on sections of a vector bundle a finite dimensional space?
10
votes
1answer
233 views

Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...
3
votes
0answers
81 views

Isometric embedding for manifolds with conical singularities?

Motivation: In the 2+1 dimensional gravity theory, solutions of Einstein equation are locally with constant curvature except at the locus of sources. In this paper the authors investigate solutions ...
3
votes
0answers
70 views

Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave?

Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave; i.e. the geodesic curvature along the boundary points ...
3
votes
0answers
36 views

Dimension of the space of Jacobi fields along $\gamma$ vanishing at $p$ and $q$ is even?

Let $G$ be a compact Lie group with a bi-invariant metric. Let $p$ be a point, and let $q$ be conjugate to $p$ along a geodesic $\gamma$. Does it necessarily follow that the dimension of the space of ...
1
vote
1answer
120 views

Zero set of eigenfunction along a sub manifold

Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
1
vote
1answer
140 views

Is there an algebraic way to characterise the ordinary integral flags?

Fix a vector space $V$ and an integer $1\leq n<\dim V$. If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ ...
8
votes
0answers
204 views

Is the formal neighborhood of the diagonal a generalization of the Jet bundle?

Let $f: X \to S$ be a morphism of locally ringed spaces and $\triangle: X \to X \times_S X$ the corresponding diagonal morphism with kernel sheaf $\mathcal{I} = \ker \triangle^{\flat}$. ...
9
votes
0answers
59 views

On the proof by Chu-Kobayashi that transformation groups are Lie groups

Chapter I of Kobayashi's Transformation Groups in Differential Geometry contains a very general theorem on transformation groups, due to Palais. I have some questions about its proof (which I attach ...
1
vote
0answers
56 views

Comparison theorem for Lambert quadrilateral

A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle. If $AOBF$ is a Lambert ...
6
votes
2answers
186 views

Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details. Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen ...
2
votes
2answers
324 views

A question about flat connection

Let $E\to X$ be a complex flat vector bundle, and say $\nabla_0$ and $\nabla_1$ are two flat connections on it. Let $p:X\times[0, 1]\to X$ denote the projection onto the first factor. Is there a way ...
0
votes
0answers
51 views

Contact and CR Examples

What is an example of a manifold such that: (A) It is both a contact manifold and a CR manifold (B) It is a contact manifold but not a CR manifold (C) It is not a contact manifold but not a CR ...
9
votes
1answer
181 views

Dimension in Whitney's theorem

There are some interesting examples of (classes of) manifolds for which I suspect that the Whitney theorem can be strenghtened. For example it is known that every (smooth) $n$-dimensional manifold can ...
0
votes
1answer
170 views

Proper actions and diffeomorphism groups

Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold? Edit: The ...
1
vote
0answers
89 views

Eigenvalues of the D'Alembertian operator

My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...
4
votes
2answers
202 views

Holonomy of a Ricci-flat affine connection

There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
1
vote
0answers
79 views

Ohsawa-Takegoshi extension theorem along snc divisor

Let $f:X\setminus D\to Y$ be a holomorphic fibre space and $X,Y$ are Kahler manifolds, where $D$ is a snc divisor on $X$. Can Ohsawa-Takegoshi extension theorem, imply that there exists a holomorphic ...
3
votes
2answers
162 views

Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$

Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally) What if it is smooth?
1
vote
0answers
105 views

horizontal lift along fibres

Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local coordinate $(s_1,...,s_d)$ of $Y$ and a local coordinate $(z_1,...,z_n)$ of ...
0
votes
0answers
73 views

Generic Identification of surfaces in a Riemannian Manifold

Let $(M,g)$ be a three dimensional Riemannian Manifold. Can we generically identify a class of embedded surfaces $\Gamma$ in $M$ such that the following property holds: There exists a distance ...
-1
votes
1answer
110 views

Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$. More precisely: I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some ...
3
votes
1answer
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
0answers
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
0answers
108 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
0answers
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
1answer
813 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
0answers
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
1answer
300 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
1answer
191 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
2answers
268 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
2answers
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
1answer
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
1answer
190 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
0answers
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 $B_1(0)\subset \mathbb{R}^{n+1}$, ...