**3**

votes

**0**answers

184 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

**0**

votes

**0**answers

62 views

### Bispinors, polyforms, bilinears and supersymmetric manifolds

Let $(V,q)$ be a regular quadratic vector space, and let us denote by $Cl(V,q)$ the corresponding Clifford algebra. Then there exists an isomorphism of $\mathbb{Z}_{2}$-graded algebras:
...

**2**

votes

**1**answer

141 views

### Closed geodesics that cross one another frequently

Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$.
$S$ has at least three simple (non-self-intersecting), closed geodesics by
a theorem of Lyusternik and Shnirel'man.
Alternatively, ...

**1**

vote

**1**answer

126 views

### Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles

As explained in:
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their ...

**2**

votes

**0**answers

103 views

### Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
...

**1**

vote

**0**answers

143 views

### Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

**3**

votes

**0**answers

49 views

### Geodesic rays in a toric variety

Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...

**4**

votes

**3**answers

265 views

### Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...

**1**

vote

**0**answers

106 views

### Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...

**1**

vote

**1**answer

123 views

### Is there a name for the function on $TTM$ swapping the 2nd and 3rd coordinates?

I'm not so good on geometry, so I fear this is a relatively basic question.
For any $N \in \mathbb{N}$, let us identify the tangent bundle of $\mathbb{R}^N$ with $\mathbb{R}^{2N}$ in the obvious ...

**0**

votes

**0**answers

53 views

### Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...

**0**

votes

**0**answers

66 views

### Ricci solitons on non-metric spaces

A Ricci soliton is a generalization of the Einstein metric and is defined on a
Riemannian manifold (M,g) by
$\mathcal{L}_v g_{ij} + 2 R_{ij} + 2λ g_{ij} = 0$,
for some constant λ, a vector field $v$ ...

**2**

votes

**1**answer

137 views

### length comparison on negatively curved surfaces

Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...

**0**

votes

**1**answer

91 views

### Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?

let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and
...

**2**

votes

**0**answers

66 views

### How to check if a manifold can be foliated by strictly convex hypersurfaces?

Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...

**1**

vote

**1**answer

108 views

### Boundary components of a subsurface

Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...

**24**

votes

**5**answers

3k views

### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

**1**

vote

**1**answer

142 views

### Upper bound of derivative of exponential map

We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp_p: T_pM\to M$ is diffeomorphism.
We know that for any $v\in T_pM$ and $w\in ...

**2**

votes

**1**answer

144 views

### Manifold with corners [closed]

Iam looking at the following situation of a manifold $Z$ with corners.
More specifically a product of a smooth manifold X with a standard $k$-simplex $\Delta^k$.
I wish to study certain formulas for ...

**1**

vote

**0**answers

153 views

### How rigid can a rigid object be in GR?

Consider a cubic lattice of space probes, with rocket motors and lasers to measure distance, and a clock to measure time. As they more from free space to the vicinity of some black hole, they try to ...

**1**

vote

**0**answers

136 views

### Characterization of the Riemann curvature tensor

Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...

**5**

votes

**0**answers

197 views

### Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for this result?
For the sake of being definite, let's say that
“citeable” means ...

**0**

votes

**0**answers

36 views

### Unitary derivative and countable set

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...

**1**

vote

**4**answers

388 views

### Tensor calculus on the frame bundle

Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...

**2**

votes

**1**answer

123 views

### Unitary representation with fixed Casimir

Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ...

**1**

vote

**1**answer

148 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

181 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

84 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

136 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

72 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

249 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

205 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

437 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

273 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

161 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

194 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

254 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

284 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

435 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

284 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

262 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

312 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

65 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

100 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

190 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

864 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

96 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

187 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

208 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

222 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 ...