**5**

votes

**1**answer

142 views

### Laplacian eigenfunction $L^p$ norms

Suppose I have a compact surface (possibly with boundary), and consider the eigenfunctions of the Laplacian, normalized so that their $L^2$ norms are $1.$ Is there some general result or conjecture on ...

**2**

votes

**1**answer

272 views

### Do lower dimensional spheres always lie on higher dimensional spheres?

At a certain stage of my research work, I require the following fact to hold true.
A surface $S$ satisfies certain conditions so that it lies on a 4-sphere in $R^{21}$ (I have used the results proved ...

**1**

vote

**0**answers

83 views

### This weaker version of CR-structure: is it studied somewhere

When I study 5-dimensional $\mathcal{N} = 1$ supersymmetry, I came across such structure as follows.
$(R, \kappa, \Phi, M)$ is an almost contact 5-manifold, such that
\begin{equation}
\kappa ...

**0**

votes

**1**answer

77 views

### Extending connections [closed]

Usually, one views the connection $\nabla$ on a vector bundle $E \to M$ as a map $\Gamma(M, E) \to \Gamma(M,T^*M) \otimes \Gamma(M,E)) \simeq \Gamma(M,T^*M\otimes E)$. One can extend this to the ...

**1**

vote

**1**answer

183 views

### Hamiltonian Isotopy class of Lagrangian Submanifold

Let $(X,\omega)$ be a symplectic manifold, $L\subset X$ be a Lagrangian submanifold, $[L]$ denotes the Hamiltonian isotopy class. How to represent $L'\in[L]$ via $L$ (for example, a graph over $L$)? ...

**0**

votes

**1**answer

146 views

### Dimension of Inverse image

Suppose we have a smooth map between two smooth manifolds $f:M→N$ such that $\dim M>\dim N$, and let $p∈N$ be a critical value. Suppose we know that $X:=f^{-1}(p)$ is a differentiable manifold (or ...

**13**

votes

**0**answers

300 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**14**

votes

**2**answers

600 views

### Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$?
Any ...

**6**

votes

**0**answers

236 views

### Geometric meaning of the black hole horizon

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...

**5**

votes

**1**answer

390 views

### Differential Geometric Aspects of Rubber Bands

What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only ...

**2**

votes

**3**answers

251 views

### Can the Einstein Field Equations be written as Difference Equations? [closed]

Does anyone know if the Einstein Field equations have ever been written as Difference Equations, and if so does that simplify anything or produce solutions not available in the usual Differential ...

**5**

votes

**1**answer

171 views

### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...

**1**

vote

**0**answers

58 views

### Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...

**12**

votes

**0**answers

239 views

### Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...

**9**

votes

**0**answers

131 views

### Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
...

**3**

votes

**0**answers

99 views

### Focal points for the exponential map and Jacobi fields

It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...

**2**

votes

**2**answers

227 views

### Holomorphic vector fields on blow-ups of CP^2

On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone ...

**3**

votes

**1**answer

139 views

### Rational homogenous spaces and symmetric spaces

What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...

**0**

votes

**1**answer

71 views

### equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...

**3**

votes

**0**answers

210 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

45 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

195 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 ...

**4**

votes

**3**answers

298 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 ...

**4**

votes

**0**answers

139 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

**1**answer

104 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 ...

**3**

votes

**3**answers

305 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.

**8**

votes

**1**answer

176 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

328 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

357 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

164 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. ...

**22**

votes

**1**answer

694 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 ...

**1**

vote

**0**answers

162 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

51 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

**1**answer

149 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

261 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

197 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 $(M,g)$ ...

**6**

votes

**1**answer

190 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

81 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

133 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} ...

**6**

votes

**2**answers

228 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

66 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

107 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

533 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

198 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

444 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

133 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

239 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

251 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

141 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. ...

**4**

votes

**1**answer

160 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 ...