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

3
votes
0answers
23 views

What is a category of “Lepagean equivalent” or “variation problem”?

I get to know about it form Mark Gotay's work An exterior differential system approach to the Cartan form, in that paper he defined the canonical Lepagean equivalent. The following is cited from it: ...
1
vote
0answers
32 views

Gaussian curvature of conformal transformations

Let $g$ be a smooth metric and $g'=e^{v}g$, where $v$ is also a smooth function. Then it is well-known that $(*) -\Delta_g u +2k_g=2 k_{g'}e^{v}$, where $k_g$ and $k_{g'}$ are the Gaussian ...
1
vote
0answers
102 views

Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?

Let $S$ be a surface in $\mathbb{R}^3$. Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear map on the ...
0
votes
0answers
31 views

Almost complex structure and intrinsic torsion.

Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$. I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$...
3
votes
0answers
33 views

Morse index of a closed minimal surface with a small disc removed

Consider the following observation: Let $c_1$ be a geodesic on the unit round sphere $S^2$ with length $2\pi-\epsilon$, where $\epsilon$ is sufficiently small. Then $c_1$ has Morse index one ...
5
votes
0answers
127 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
2
votes
0answers
44 views

One-parameter group of nonvanishing vector field

Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$. Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
4
votes
0answers
39 views

Foliation of cylinders by constant mean curvature spheres

Let the cylinder $S^{n-1}\times \mathbb R$ be equipped with the standard metric $g$. Suppose that there exists a sequence of metrics $g_{\epsilon}$ on $S^{n-1}\times \mathbb R$ such that $g_{\epsilon}$...
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votes
0answers
34 views

Differential equation hard example [on hold]

solve the differential equation $$ (d{y}/d{x})^{3} + d{y}/d{x} = x $$ Thank you.
4
votes
0answers
92 views

Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
6
votes
0answers
165 views

The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
2
votes
0answers
37 views

Partial Liouville equation

In my master's thesis, I worked on mathematical multi-scale models for muscle tissue. Now after finishing it, I would like to find out if one direction could be a research topic for my PhD. At one ...
2
votes
0answers
153 views

Torsion free-ness of cohomology of moduli of vector bundles

My question requires a little introduction: $\textbf{Atiyah-Bott's solution:}$ In the paper "The Yang mills equation of Riemann surfaces" Atiyah-Bott has computed the cohomology of moduli vector ...
0
votes
0answers
49 views

Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form. We have a type I singularity if $$ \max_{p \in M} |A(p,...
5
votes
0answers
46 views

Eigenvalue lower bounds for manifold with positive Ricci curvature

For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...
2
votes
0answers
100 views

Gauss Bonnet theorem calculation for pseudosphere

In an attempt to verify Gauss-Bonnet theorem for a Beltrami pseudosphere I calculated a simple case of the Riemann sphere. Taking curved radius $a$ for geodesic polar co-ordinate from the smooth ...
6
votes
0answers
72 views

Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently, $$A\natural B =(BA^{-1})^{1/2}A=A(A^...
2
votes
1answer
191 views

An orientable compact even dimensional manifolds whose all even cohomologies do not vanish but it does not admit any symplectic structure

What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any ...
0
votes
1answer
96 views

Metric of non-negative scalar curvature

Let $(M,g)$ be a closed Riemannian manifold, if $Scal^g>0$, we know that in a metric space of $M$, there is a neighborhood of $g$, such that all metrics in this neighborhood have the positive ...
5
votes
1answer
125 views

Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces

I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...
2
votes
1answer
63 views

Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation

Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding $$ |\mathbf{E}_{X, Y\sim P \...
2
votes
0answers
44 views

Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)

Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...
1
vote
0answers
58 views

Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary

In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669 Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
3
votes
1answer
154 views

Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$

Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$? I have already discussed this question in the following post ...
5
votes
0answers
212 views

Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
3
votes
0answers
103 views

Differential operators on a compact Lie group associated to bracket-generating sets

Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$. Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$. Assume that $\{X_1,\dots,X_h\}$ is ...
3
votes
1answer
93 views

Property of distance function with “smoothly” varying Riemannian metrics

Let $(M,g)$ be a smooth and compact Riemannian manifold. Suppose I have a "smoothly varying" (precisely formulating this is part of the question) one parameter family of Riemannian metrics $(M,g_t)$ ...
3
votes
1answer
116 views

Cheeger inequality for measures

Given a probability measure $\mu$ on $\mathbb{R}^n$, its Poincare constant is the least number $C$ such that: $$ \int f^2 d\mu \leq C\int \|\nabla f\|^2 d\mu $$ for all zero mean function $f$. Is ...
3
votes
0answers
126 views

Analytic Aspects of Rational Maps

I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
2
votes
0answers
75 views

The complex Clifford algebra

If $(E,g,w)$ is a vector space $E$ with a metric $g$ and a symplectic form $w$; then we can define the complex parts $(1,0)$ and $(0,1)$, so that the complex Clifford algebra is: $$e_1 . f_1 + f_1 . ...
1
vote
1answer
48 views

Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...
4
votes
2answers
214 views

Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs, where $P_k$ is given by \begin{equation} \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
7
votes
2answers
262 views

Nash isometric embedding for noncompact manifolds

It seems that the smooth isometric embedding theorem by Nash is true also for noncompact manifolds. Is it true that any (complete, connected) Riemannian manifold $(M^n,g)$ admits a proper smooth ...
4
votes
4answers
346 views

Motivation for construction of Associated fiber bundle from a principal bundle

Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)...
0
votes
0answers
65 views

Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...
2
votes
0answers
110 views

A problem on real analysis related to Hilbert's fourth problem

This is an extensive re-write of a question I deleted and which had basically the same title. Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by ...
3
votes
3answers
351 views

Alternative (easier) Proof of Ambrose Singer Holonomy theorem

Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...
0
votes
0answers
49 views

Clarification needed on vector field conditions in Smale's “On gradient dynamical systems”

I previously posted the question on MSE but I haven't received an answer. I'm now posting it here in a slightly revised form. In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. ...
3
votes
2answers
292 views

Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
3
votes
0answers
204 views

Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
4
votes
0answers
85 views

Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
3
votes
0answers
63 views

Existence of harmonic symplectic structure on symplectic Riemannian manifold

This post is an expanded version of this MSE post. Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric. Is there a symplectic structure $\...
2
votes
0answers
96 views

Are $C^1$ immersions dense in $C^1$?

Let $M$ be a closed compact manifold. Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
15
votes
0answers
384 views

What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
3
votes
2answers
99 views

Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology ...
6
votes
1answer
197 views

is signed distance function real analytic for real analytic domains

If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by \begin{equation} f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\...
3
votes
1answer
47 views

Cut on hypersurfaces and angular defects

I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of ...
6
votes
1answer
307 views

Manifolds with negative dimension – Definition, References

Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...
4
votes
1answer
90 views

Monotonicity of infimum of the Willmore energy with prescribed genus

Let $$ \beta_g:=\inf\{\frac14\int_\Sigma H^2 d\mu \hspace{0.2cm} | \hspace{0.2cm} \Sigma\subset \mathbb R^{3}, \operatorname{genus}(\Sigma)=g \} $$ be the infimum of the Willmore energy of embedded ...
1
vote
0answers
43 views

Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators? In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...