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

learn more… | top users | synonyms (1)

0
votes
0answers
39 views

Expansion of the squared distance function from a submanifold

let $(M,g)$ be a compact Kahler manifold of dimension $m$ with the Kahler metric $g$ real-analytic. Let $N\subset M$ be a complex submanifold of dimension $0<k<m$. Let $$d_{g}(M,\cdot): ...
3
votes
1answer
139 views

Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?

Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$. Assume that $S$ satisfies ...
0
votes
0answers
74 views

Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form: Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...
1
vote
2answers
106 views

Right invariant Killing fields of Right invariant Riemanian metrics

Can there exist a right invariant killing field of a right invariant (but not bi-invariant) Riemannian metric on a Lie group? I am especially interested in the case of $SU(N)$ with a metric of the ...
4
votes
1answer
489 views

The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...
7
votes
0answers
228 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
3
votes
1answer
172 views

Displaceability of submanifolds

My question is motivated by the following question. How transitive are the actions of symplectomorphism groups ? A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there ...
1
vote
0answers
50 views

Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting $$ (\exp_x^* ...
0
votes
1answer
75 views

Optimal Regularity for Invariance of Curvature under Isometries

It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...
12
votes
2answers
881 views

Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...
1
vote
0answers
39 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold. Explanation Suppose we have a 3D smooth manifold ...
2
votes
0answers
65 views

Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine ...
3
votes
1answer
250 views

Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary. Is there a ...
0
votes
0answers
281 views

A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version. Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...
1
vote
1answer
247 views

Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $LG(2,4)$?

On the paper http://arxiv.org/abs/1009.1364 (published on Proc. London Math. Soc.) I've found an interesting statement: The Lie quadric $Q^3$, i.e., the space of all points, lines and circles ...
0
votes
1answer
193 views

A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$ is integral, for some almost complex structure ...
1
vote
0answers
61 views

coordinate system in AF manifold

we know that an end of $(M,g)$ is asymptotically flat if it is diffeomorphic to the complement of a compact set $K$ in $\mathbb{R}^3$, and the metric tensor $g$ satisfies ...
3
votes
2answers
235 views

The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group? (My ...
0
votes
1answer
301 views

How to prove this Weitzenbock formula?

In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, ...
0
votes
0answers
43 views

Special family of Metrics on Transitive Lie Algebroids?

Let $\rho:E\longrightarrow TM$ is a transitive Lie Algebroid, then $L=ker\rho$ is bundle of lie algebras. Suppose $\Gamma:TM\longrightarrow E$ be a linear splitting. Define $$\nabla_X ...
1
vote
1answer
109 views

Which book will discuss torsion tensor and affine connection in detail? [closed]

Which book about differential geometry will have these formula about torsion tensor? $$\nabla_{j}T^{i}_{kl}+\nabla_{k}T^{i}_{lj}+\nabla_{l}T^{i}_{jk}=R^{i}_{jkl}+R^{i}_{klj}+R^{i}_{ljk}$$ ...
3
votes
1answer
151 views

Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...
0
votes
0answers
100 views

positive eigenfunction on complete Riemannian manifold

Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation $$ \Delta u = u $$ where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...
9
votes
2answers
437 views

Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...
2
votes
0answers
317 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
1
vote
0answers
49 views

Allen Cahn Equation with Dirichlet Data

consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions ...
1
vote
1answer
369 views

A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way. ...
1
vote
1answer
109 views

metric has morse index 2

I am reading Richard Schoen's classical example on the multiplicity of solutions of yamabe problem. He says on $S^1(T)\times S^{n-1}$, there exists a critical number $T_0$ such that if $T\leq T_0$, ...
9
votes
1answer
490 views

On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result: A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm. To be clear about ...
5
votes
0answers
151 views

Non-trivial isometric embedding of the standard sphere into $\mathbb{R}^n$? [closed]

Let $S$ be the unit sphere in $\mathbb{R}^3$. Is it possible to embed $S$ isometrically into some $\mathbb{R}^n$, $n>3$, such that the image does not lie on any 3-dimensional affine subspace? More ...
3
votes
1answer
133 views

All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...
6
votes
0answers
133 views

Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
2
votes
0answers
177 views

Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact $$ ...
0
votes
0answers
72 views

Comparison of support of a divisor and its class in de Rham cohomology

Let $X$ be a smooth surface over $\mathbb{C}$ and $D$ be an effective divisor. Suppose that the linear system corresponding to $D$ is zero dimensional. Denote by $[D] \in H^2(X,\mathbb{Z})$ the ...
2
votes
0answers
102 views

Notions of convexity for the boundary of a Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. Question: What does it mean to say that the boundary is convex and strictly convex? Alternatively: What ...
2
votes
0answers
226 views

Existence of diagonalizing coordinates for the metric tensor

Solving for metrics that are Einstein, i.e that satisfy $R_{\mu \nu} = \Lambda g_{\mu \nu}$ is highly non-trivial as soon as $g_{\alpha \beta}$ is allowed to have off-diagonal components. However, ...
7
votes
1answer
226 views

How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map $$ \mu: G\times ...
15
votes
1answer
606 views

When is the Gromov--Hausdorff limit of a sequence of manifolds itself a manifold?

Suppose a metric space $X$ is the Gromov--Hausdorff limit of a sequence of Riemannian three-manifolds $M_i$ with Ricci curvature bounded from below and not collapsed. Is $X$ a manifold? What ...
14
votes
3answers
296 views

Existence of sections of the evaluation map for the diffeomorphism group

Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point ...
5
votes
3answers
493 views

Non-continuous differentiability for differential forms

Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior ...
2
votes
0answers
115 views

A question on Hilbert geometries as metric-measure spaces

Recall that a Hilbert geometry is the interior of a convex body $K \subset \mathbb{R}^n$ provided with the metric $$ d(x,y) = \frac{1}{2} \ln\left(\frac{|x-b|}{|y-b|}\frac{|y-a|}{|x-a|}\right) , $$ ...
1
vote
1answer
146 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
4
votes
2answers
471 views

A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...
1
vote
1answer
202 views

A special type of transitivity

Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$. Define $B$=All linear volume ...
1
vote
1answer
122 views

Vertices of Curves and Eigenvectors of Hessian

This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$. This curve is strictly convex, and feel free to ...
1
vote
0answers
49 views

Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometry

Let $M$ be an open Riemannian surface of bounded geometry. Let $\Gamma$ be a group of diffeomorphisms of $M$. Suppose that $\Gamma$ is equi-quasi-isometric; i.e., its elements are (differentiable) ...
1
vote
0answers
103 views

infranilmanifolds: harmonic forms parallel?

I am studying Lott's paper : "On the spectrum of a finite volume negatively-curved manifold" and the satement is following: We have an compact infranilmanifold $N$ which is finitely covered by a ...
3
votes
1answer
281 views

Does the cohomology after Dehn surgery depend only on the original 3-manifold or also how the knot is situated?

For $f:S^1\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a $0/1$-type Dehn surgery along $f$: First remove from $M$ a solid torus which is a tubular neighbourhood of the knot ...
4
votes
1answer
253 views

Derivation of yamabe flow

I am reading papers about yamabe flow. I have a problem about how people derive it as a gradient flow. Suppose we have $(M,g_0)$, $g(t)=u^{\frac{4}{n-2}}(t)g_0$ is another conformal metric. Let ...
2
votes
2answers
212 views

vector field on a curve as projection of a constant vector field on an embedding space

Suppose a tangent vector field is given on a planar curve and one asks the following question: What is the condition on this tangent vector field that it comes from the ordinary Euclidean projection ...