**1**

vote

**0**answers

99 views

### A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles.
For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...

**1**

vote

**1**answer

29 views

### Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...

**0**

votes

**1**answer

174 views

### First chern class of fibers of compact Kaehler algebraic variety

Let $M$ be an compact Kähler algebraic variety and suppose $K_M$ is semi-ample. Consider the holomorphic map $\pi:X\to \Sigma \subset \mathbb CP^N$ with $Kod(M)=dim_\mathbb C\Sigma$ (here $Kod$ means ...

**3**

votes

**3**answers

235 views

### Generic absence of non-trivial first integrals of geodesic flows

Is it true that given a smooth manifold M (with or without boundary), a "generic" metric g on M does not possess any non-trivial (non-constant) first integral for the geodesic flow induced by g on the ...

**2**

votes

**0**answers

97 views

### On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...

**1**

vote

**0**answers

71 views

### Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...

**1**

vote

**1**answer

168 views

### Canonical Metric on Grassmann Manifold

I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject.

**6**

votes

**0**answers

106 views

### Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...

**2**

votes

**2**answers

289 views

### All Kähler metrics on a complex manifold?

Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$?
For the case of surfaces ($dim_C=1$), ...

**4**

votes

**1**answer

276 views

### Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?

Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...

**4**

votes

**0**answers

143 views

### Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...

**21**

votes

**1**answer

622 views

### What is the analogue of simple prime closed geodesic for prime numbers?

The prime geodesic theorem (of Margulis?) states that on a compact surface of (constant?) negative curvature, the number of prime closed geodesics of length at most $L = \log x$ is approximately ...

**0**

votes

**0**answers

122 views

### connections and curvature

Let $(M, g)$ be a Riemannian manifold. Is it possible to construct two different affine (or metric) connections, say $\nabla$ and $\nabla'$, which induce the SAME curvature tensor, i.e. $R(X, ...

**2**

votes

**0**answers

46 views

### Almost sure transversality of smooth random maps

I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO.
Let me first formulate the ...

**1**

vote

**0**answers

79 views

### Homological properties with discontinuity

I am trying to find information about modeling discontinuous fields over a manifold.
In particular, I have a manifold that may change its homology class, for instance I can have a growing circular ...

**1**

vote

**1**answer

120 views

### Derivability properties of the distance function in a Finsler Manifold

We know that, in a Riemannian manifold, the geodesic distance between a point O and a point P, when we fix O, is a function of P that is $C^\infty$ everywhere on a local neighborhood, except in P=O. ...

**5**

votes

**2**answers

243 views

### Affine structures

I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy ...

**4**

votes

**1**answer

161 views

### Causal+Imprisonment Condition+Compact Clausure -> Stably Causal?

My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (http://projecteuclid.org/euclid.cmp/1103899983) One of the results there is:
Let $(M,g)$ ...

**5**

votes

**1**answer

146 views

### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...

**0**

votes

**0**answers

77 views

### What are the most important papers to read about Ricci flow? [duplicate]

I would like to know, which papers are important to read, if one wants to work about Ricci flow. What papers have to be known by every person conducting research in this field?
Can you provide me with ...

**1**

vote

**1**answer

118 views

### Euler-Poincare equations with constraints

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ...

**2**

votes

**1**answer

137 views

### Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group:
$$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$
Here, $H$ is an NxN skew-Hermitian matrix (for very ...

**2**

votes

**1**answer

61 views

### Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...

**0**

votes

**0**answers

90 views

### Existence of the Dirichlet heat kernel for arbitrary open subsets?

consider first of all an open and bounded subset $\Omega\subset\mathbb{R}^n$, s.t. the boundary $\partial \Omega$ is a manifold of class $C^2$. Then I know that there exists a Dirichlet heat kernel, ...

**12**

votes

**2**answers

505 views

### Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in ...

**2**

votes

**1**answer

189 views

### The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...

**10**

votes

**2**answers

417 views

### Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as ...

**1**

vote

**1**answer

115 views

### Equivalent Killing vector fields via an isometry

Suppose that $(M,g)$ is a complete semi-Riemannian manifold.
We say that two Killing vector fields $V$ and $W$ are equivalent if there is $\Phi:M\rightarrow M$ an isometry such that $\Phi_*(V)=W$.
...

**4**

votes

**1**answer

225 views

### Spectral multipliers vis-a-vis Differential geometry

Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...

**2**

votes

**1**answer

200 views

### Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...

**4**

votes

**2**answers

125 views

### Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...

**3**

votes

**1**answer

143 views

### Vector Laplace Beltrami operator of the Gauss map

Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami ...

**0**

votes

**1**answer

253 views

### Yang-Mills equations are not elliptic [closed]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic?
Alternatively, how does one calculate the principal symbol of the Yang-Mills equations?
Can ...

**11**

votes

**0**answers

302 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**1**

vote

**0**answers

73 views

### Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...

**1**

vote

**0**answers

185 views

### Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...

**1**

vote

**1**answer

222 views

### Example of a specific manifold

I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat.
Does there exists such manifolds?
Thanks.

**5**

votes

**0**answers

84 views

### Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...

**0**

votes

**1**answer

226 views

### Computing the Chern class of $S^6$ [closed]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...

**3**

votes

**2**answers

196 views

### Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...

**7**

votes

**3**answers

564 views

### nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...

**14**

votes

**2**answers

726 views

### If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...

**1**

vote

**0**answers

78 views

### why is this result about Gaussian analytic functions equivalent to the Crofton formula

I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function.
Definition A Gaussian analytic function ...

**6**

votes

**1**answer

319 views

### Complex geometry text/research introduction for the analyst

To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...

**4**

votes

**1**answer

69 views

### Point of maximal distance on a non-positively curved PL surface

I just posted this question as a comment to the question Hypersurfaces and Elliptic Points
but I don't know how many people will see it.
It's well known and easy to prove that a point on a closed ...

**0**

votes

**0**answers

56 views

### the push forward of the differential idea of sheaf

This is about the differential idea of a sheaf as a vector bundle $E$ (not necessarily locally trivial) on a real manifold $M$ with a zero curvature connection $\nabla$. The zeroth cohomology is then ...

**3**

votes

**1**answer

280 views

### Mathematica package for supergravity and string theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...

**0**

votes

**0**answers

85 views

### Estimate of Mean curvature

If $\varphi: \Sigma\hookrightarrow (\mathbb{R}^{n+1} ,g)$ is an embedded convex hypersurface, can we estimate the mean curvature growth or integral in terms of intrinsic geometry, such as the ...

**2**

votes

**1**answer

133 views

### Norms on Clifford algebra (C^* norm)

Basically I'm interested in operator algebras such as $C^*$ or von Neumann algebras. However I decided to learn a bit about noncommutative geometry (in particular spectral triples). Before doing this ...

**5**

votes

**1**answer

285 views

### The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and ...