**4**

votes

**0**answers

38 views

### Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a “Wick rotation”?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like
...

**3**

votes

**1**answer

69 views

### Action generated by geodesic flow is hamiltonian

I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle ...

**5**

votes

**2**answers

500 views

### flat metrics on the 2-sphere with conical singularities

Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p_1,\ldots,p_n$.
What is the moduli space of such metrics up to isometry? In particular what is its dimension?

**72**

votes

**8**answers

8k views

### What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...

**3**

votes

**2**answers

181 views

### Triangles in rigid Riemann surfaces

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows:
Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...

**6**

votes

**1**answer

292 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**10**

votes

**1**answer

753 views

### Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...

**0**

votes

**1**answer

107 views

### Symmetries of non-Riemannian curvature tensor

The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...

**4**

votes

**3**answers

350 views

### Do cotangent bundles have “bounded geometry”?

I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and ...

**10**

votes

**2**answers

434 views

### Witten's proof of Morse inequalities, question on eigenvalues?

See here. I present Theorem 6 and Corollary 7 as follows.
Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.
Corollary 7. $\dim ...

**2**

votes

**1**answer

64 views

### Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...

**4**

votes

**2**answers

270 views

### What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...

**4**

votes

**2**answers

139 views

### Are periodic billiard trajectories stable on a manifold with strictly convex boundary?

Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reclects specularly at the boundary).
...

**-1**

votes

**0**answers

32 views

### Angle sum of triangle in Schwarzschild solution [on hold]

Curvature of space is often intuitively explained as angles of a triangle not adding up to 180 degrees. I was wondering how well that applies in the context of General Relativity.
Suppose you have a ...

**4**

votes

**2**answers

454 views

### Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...

**6**

votes

**1**answer

141 views

### Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be ...

**10**

votes

**2**answers

890 views

### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

**0**

votes

**0**answers

18 views

### c-superdifferential is unique +cost function is differentiable, then the potential function is differentiable?

Let $M$ be a compact Riemannian manifold, $\mu$ and $\nu$ are two Borel probability measures, the cost function $c(x,y)=\frac{d^2(x,y)}{2}$.
It's well known that the infimum of the Kontorovich's ...

**3**

votes

**1**answer

188 views

### Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...

**4**

votes

**0**answers

101 views

### Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set ...

**23**

votes

**3**answers

1k 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 ...

**9**

votes

**1**answer

365 views

### Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?

**1**

vote

**0**answers

78 views

### Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads:
If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...

**3**

votes

**1**answer

290 views

### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

**4**

votes

**1**answer

161 views

### Check symplectomorphism property on infinitesimal generators

I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G ...

**1**

vote

**1**answer

366 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

**3**

votes

**1**answer

322 views

### Survey papers on the role played by PDE in mathematics

There are already several questions on Mathoverflow about the application of PDE to several other topics (e.g., algebraic and differential geometry and topology, number theory, harmonic analysis, ...

**8**

votes

**1**answer

369 views

### Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...

**0**

votes

**0**answers

44 views

### Stabilizer subgroup in adjoint action [migrated]

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...

**13**

votes

**3**answers

1k views

### Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when ...

**9**

votes

**2**answers

269 views

### References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials?
In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...

**52**

votes

**2**answers

4k views

### Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem:
Complex structure on the six dimensional sphere from a spontaneous symmetry breaking
Journ. Math. Phys. 56, ...

**0**

votes

**0**answers

57 views

### Petrov type D spacetime

I have a rather general question about the Petrov type D spacetime.
The standard definition (also used in the official Wikipedia article) is that the Petrov classification is according to which Weyl ...

**2**

votes

**1**answer

136 views

### Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
The background
So he wants to show that any symplectic form is ...

**4**

votes

**0**answers

118 views

### Explicit metrics on non-compact Calabi-Yau threefolds

I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known.
For instance, an important class of such spaces can be constructed algebraically, including local ...

**4**

votes

**0**answers

53 views

### Space of piecewise geodesic paths

It is well-known that for any Riemannian manifold $M$, the spaces
$$H_x(M) = \Bigl\{ \gamma \in AC\big([0, 1], M\big) ~\Big|~ \gamma(0) = x, \int_0^T|\dot{\gamma}(s)|^2 \mathrm{d}s < ...

**0**

votes

**0**answers

76 views

### Poincare inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...

**2**

votes

**1**answer

91 views

### Partial differential equation from Kirchhoff system

I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$
$$\partial_t \vec{u} = \partial_s \vec{w} - [\vec{w} \times \vec{u}].$$
Using ...

**1**

vote

**0**answers

329 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics ...

**2**

votes

**1**answer

171 views

### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...

**2**

votes

**1**answer

342 views

### Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context:
I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...

**1**

vote

**0**answers

114 views

### Symplectic spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry.
The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le ...

**0**

votes

**0**answers

135 views

### How to change the given metric if we want to add few extra isometries?

I have a Hilbert Space $X$ and a group $G$, which consists of bounded linear self-bijections of $X$ (if it helps, this group has a locally compact, but not compact topology). Is there a canonical way ...

**5**

votes

**2**answers

165 views

### Permutable (Lie) subgroups

Let's recall that, a group $G$ being given,
two subgroups $A,B\subset G$ are called
permutable iff $AB=BA$ for the Minkowski
law. It is straightforward to see that $(A,B)$
are permutable iff $AB$ ...

**3**

votes

**1**answer

318 views

### Einstein metric for simply connected Riemannian manifold, diffeomorphic to 3-sphere?

If we can construct an Einstein metric for a simply connected Riemannian manifold, why does it necessarily follow it is diffeomorphic to the $3$-sphere?

**2**

votes

**0**answers

281 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 ...

**8**

votes

**1**answer

222 views

### What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$?
The $p$-Laplacian ...

**1**

vote

**0**answers

119 views

### Oriented volume and determinants: Circularity [duplicate]

One often reads about oriented volumes as a motivation for determinants. In two dimensions you can scetch some nice pictures which may convince students that it is a good idea to have a closer look at ...

**5**

votes

**1**answer

113 views

### Isotropic subvarieties of $ V(x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2) $

Let $ K $ be a field, $ \operatorname{char} K = 0, $ let $ Q = x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2 $ be the totally isotropic form, then the maximal linear isotropic
subspaces of $ V(Q) \subset ...

**6**

votes

**0**answers

91 views

### Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...