Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
0
votes
0answers
27 views
Integral geometry and curvature of surfaces
I'll stick to 2-surfaces in $\mathbb{R}^3$ for simplicity. Higher dimensions generalizations welcome.
In classical integral geometry, we may obtain up to a scaling factor, for example, the surface ...
3
votes
1answer
30 views
Foliations of Lorentzian manifolds by Spacelike Hypersurfaces
Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
0
votes
0answers
56 views
Rolling map as a diffeomorphism?
Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map)
$$(\Phi\gamma)(t) = ...
8
votes
1answer
169 views
Piecewise linear (PL) structures on $\mathbf R^4$
One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...
0
votes
0answers
88 views
On Gromov's Theorem on Symplectic Homotopy
I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
4
votes
2answers
371 views
Based loop groups as stacks?
I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension ...
3
votes
1answer
165 views
Existence and uniqueness of a quasi-linear pde system on a surface
I have the following system of first order quasi-linear pde:
$$ -(\Delta+1) a^{\alpha\beta} [b_{\beta\rho} I_{\alpha;\sigma}+b_{\beta\sigma} I_{\alpha;\rho}]
+ a^{\alpha\beta} [(\Delta+1) ...
-1
votes
1answer
155 views
Reductive space & Reductive Lie algebra
If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...
-1
votes
0answers
48 views
Intuition for Killing vectors in negative-definite Ricci tensor
Theorem 4.3 from Chapter II of Kobayashi's Transformation Groups in Differential Geometry states that, if $(M,\mathrm{g})$ is a Riemannian manifold with negative definite Ricci tensor, then any ...
2
votes
1answer
171 views
Do all surfaces (2d riemanian manifolds) admit constant curvature? [closed]
There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a ...
2
votes
1answer
185 views
Boundary geometry of a contact manifold
Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact ...
5
votes
1answer
251 views
Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
-2
votes
0answers
57 views
Surface and Laplace's equation [closed]
I'm doing a study of surfaces with constante curvature and I'm trying to solve this equation:
$$\Delta\phi = -e^{2\phi}K_0$$
for a 2-dimensional metric with constant Gauss curvature such that rotation ...
17
votes
1answer
384 views
Super-cobordisms
One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
5
votes
2answers
242 views
Alternative proof of Varadhan's formula on Riemann manifolds
Consider Varadhan's famous formula for the kernel of the heat equation on a manifold:
$$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$
I do not have access to his 1967 two ...
2
votes
1answer
214 views
Triviality of holomorphic vector bundles over contractible Stein manifolds
If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
1
vote
2answers
214 views
Line bundles over Kahler-Hodge manifolds
A Kahler-Hodge manifold $M$ can be defined as a Kahler manifold whose Kahler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists an hermitian ...
0
votes
1answer
57 views
Fundamental solution to the heat equation with zero boundary values
let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ ...
0
votes
0answers
93 views
Continuous isometries on Ricci flat compact manifolds
If I am not mistaken, a compact Ricci-flat manifold can have at most torus isometries. What is the name of the corresponding theorem or where can I find this result proven?
It is known that ...
1
vote
1answer
170 views
Linearisation of Einstein operator
Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The Ricci curvature can be viewed as a differential operator ...
20
votes
1answer
639 views
fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
1
vote
1answer
84 views
A question about horizontal lifts for an Ehresmann connection
I was just reading the Ehresmann connection wikipedia page and noticed that it defines an Ehresmann connection to be complete if a curve in the base can be horizontally lifted over its entire domain. ...
4
votes
1answer
163 views
The heat kernel as an exponential of an integral
In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:
$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, ...
2
votes
1answer
186 views
An identity for Futaki-Donaldson invariant
Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...
2
votes
2answers
210 views
Length of non-horizontal curve
Let $M$ be a sub-Riemannian space.
Consider a smooth curve $\gamma:[0,1]\to M$ such that
$\dot\gamma(t)\not\in H_{\gamma(t)}$, where $H_{\gamma(t)}$ is the horizontal subbundle ( i.e. $\gamma$ is ...
2
votes
2answers
156 views
Complex manifolds with trivial canonical bundle
It is known that a compact Calabi-Yau manifold can be defined as a compact Kahler manifold $M$ with trivial canonical bundle, or alternatively, a reduction of the structure group from $U(n)$ to ...
6
votes
0answers
175 views
The open problem of finding the explicit metric on a compact Calabi-Yau manifold
If I am not mistaken, no explicit metric on a compact Calabi-Yau manifold is known. I guess part of the difficulty is due to the fact that compact Calabi-Yau manifolds do not admit continuous ...
1
vote
1answer
144 views
Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a Kahler-Hodge manifold? (possible open problem)
I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer. The problem is:
Is the manifold
...
4
votes
1answer
173 views
dual of the Lie derivative
Let $\Omega^p(M)$ be the smooth degree $p$ differential forms on an $n$-dimensional manifold $M$. The Hodge $\ast$ operator maps $\ast : \Omega^p(M) \to \Omega^{n-p}(M)$. Using the Hodge dual we can ...
4
votes
1answer
213 views
Obstruction to a $SU(4)$-structure in eight dimensions
What is the obstruction for the existence of a $SU(4)$-structure on a spin, eight-dimensional manifold $M$? This is equivalent to the existence of two nowhere vanishing global sections of the ...
1
vote
1answer
79 views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain ...
2
votes
3answers
192 views
Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case
Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write
$$\dim H^0(X,L^k)=a_0k^n+a_1k^{n-1}+...$$
If $X$ is smooth then $a_0=Vol(X)$ and we can compute $a_i$.
If $X$ is ...
3
votes
0answers
144 views
“Parallel translate” of a geodesic in the following sense [closed]
Since I'm lazy, I'm shamelessly referring to the following question:
Derivative of Exponential Map
Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector ...
2
votes
1answer
102 views
Locally conformal Kahler manifolds with SU(4) structure
I would like to know if there exist eight-dimensional manifolds such that:
It has SU(4)-structure.
It is locally conformal Kahler.
It is not a Calabi-Yau four-fold.
A weaker question that also ...
3
votes
1answer
100 views
Distance function from a topological submanifold
Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can ...
1
vote
1answer
115 views
How to find isothermal coordinates equivalent to circles in far limit?
I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...
2
votes
1answer
135 views
Lorentzian metrics on the torus up to continuos deformations
Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.
Which ...
-1
votes
0answers
146 views
Flat vector bundles and constant transition functions [migrated]
Let $E\to M$ be a vector bundle endowed with a flat connection. Then, does $E$ admit a bundle atlas with constant transition functions?
For a vector bundle with constant transition functions, are ...
15
votes
1answer
347 views
Integrals of pullbacks and the Inverse function theorem(s?)
The usual story goes like this:
Smooth picture (?):
For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
is true:
$\phi^{-1}$ is a local diffeomorphism a.e.
...
0
votes
0answers
67 views
Integration over a second order tensor [migrated]
I would like to compute the mean value of a second order tensor $\mathbf{T}$
expressed in planar cylindrical coordinates.
The mean value for any second order tensor is (reference [1] page 101)
...
1
vote
0answers
117 views
Dirac operator in Generalized Geometry
I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...
4
votes
0answers
78 views
Infinitesimal Generator of Billiard Flow
The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
4
votes
0answers
249 views
Obstructions to deformations of complex manifolds
Roughly, a deformation of a compact complex manifold $M$ (in the sense of Kodaira-Spencer) is a triple $(\mathcal{M},w,B)$ where $w:\mathcal{M}\to B$ is a holomorphic map over domain $0\in B\subset ...
0
votes
0answers
99 views
Generalization of the Riemann curvature tensor
The Riemannian curvature tensor (also holding for manifolds with torsion) is for the vector fields $X,Y,Z$ formally given by: $R(X,Y)Z=(∇ X ∇ Y −∇ Y ∇ X −∇ [X,Y] )Z$ .
This tensor clearly exist for ...
-3
votes
1answer
141 views
Fibre bundles and flat connections [closed]
If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat ...
0
votes
0answers
46 views
definition of derivative of a vector field [migrated]
i try to define by myself the notion of differentiation of a vector field on a general manifold.
I know that it is a classical subject and that there exist some answers as Lie derivative of a vector ...
0
votes
0answers
60 views
Wang's C-subgroups and M-manifolds
Let $K$ be a semisimple compact Lie group.
In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer ...
3
votes
1answer
128 views
invariant measure of uniquely ergodic horocycle flow
Let $S$ be a compact connected orientable surface of variable negative curvature, and let $M=T^1S$ be the unit tangent bundle of $S$. Then, we know from the paper of Brian Marcus (*) that the negative ...
0
votes
0answers
168 views
When an Spherical variety is $K$-stable
Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When an Spherical variety is $K$-stable? Is ...
11
votes
1answer
341 views
Schemes over topological rings
I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...
