Tagged Questions
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 $g_{\epsilon}$...
-6
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 \...
