Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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4
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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
1answer
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
2answers
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
8answers
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
2answers
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
1answer
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
1answer
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
1answer
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
3answers
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
2answers
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
1answer
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
2answers
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
2answers
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
0answers
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
2answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
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1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
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0answers
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 ...