**1**

vote

**1**answer

150 views

### Are there compact Riemannian manifolds whith Q-curvature negative?

Are there known examples of compact Riemannian manifolds with Q-curvature negative?

**2**

votes

**2**answers

182 views

### boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$

Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...

**0**

votes

**1**answer

87 views

### How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...

**5**

votes

**1**answer

145 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

**0**answers

75 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) = ...

**10**

votes

**1**answer

251 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

**1**answer

222 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

**2**answers

447 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

**1**answer

280 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

**1**answer

165 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, ...

**2**

votes

**1**answer

197 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 ...

**3**

votes

**2**answers

264 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

**1**answer

287 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 $M- \partial M$. ...

**17**

votes

**1**answer

440 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

**2**answers

304 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

**1**answer

272 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

**2**answers

318 views

### Line bundles over Kähler–Hodge manifolds

A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...

**0**

votes

**1**answer

72 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

**0**answers

106 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

**1**answer

192 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 ...

**22**

votes

**2**answers

872 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

**1**answer

102 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

**1**answer

201 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

**1**answer

232 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

**2**answers

230 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

**2**answers

201 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

**0**answers

205 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 ...

**3**

votes

**1**answer

362 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? 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.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times ...

**4**

votes

**1**answer

195 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

**1**answer

228 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

**1**answer

91 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 ...

**3**

votes

**3**answers

217 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

**0**answers

166 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

**1**answer

122 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 ...

**4**

votes

**1**answer

119 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

**1**answer

130 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

**1**answer

151 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 ...

**15**

votes

**1**answer

387 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.
...

**1**

vote

**0**answers

133 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

**0**answers

92 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

**0**answers

279 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

**0**answers

119 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

**1**answer

175 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

**0**answers

67 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

**1**answer

172 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 ...

**1**

vote

**0**answers

233 views

### When a 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 a Spherical variety is $K$-stable? Is ...

**11**

votes

**1**answer

362 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 ...

**1**

vote

**2**answers

187 views

### Defining Gauss-Kronecker curvature for submanifolds of $\Bbb R^n$ and relation with ${\rm d}{\bf N}_i$'s

I'm trying to find a definition for Gauss-Kronecker curvature of submanifolds of $\Bbb R^n$, but I'm only finding it for hypersurfaces. I would like to know if someone knows any text which works in ...

**1**

vote

**0**answers

121 views

### Hermitian metric on conic Kaehler-Einstein setting

I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler ...

**0**

votes

**1**answer

128 views

### Gluing submanifolds along their common boundary

This might be too elementary for this site, but I asked first on math.stackexchange and didn't get an answer even after offering 250 bounty points.
Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ ...