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

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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
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1answer
139 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
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2answers
252 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
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1answer
193 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
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1answer
157 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 ...
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0answers
78 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 ...
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1answer
130 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} ...
3
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1answer
157 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 ...
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1answer
64 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 ...
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0answers
95 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, ...
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2answers
518 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
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1answer
192 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
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2answers
430 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 ...
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1answer
117 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
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1answer
231 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 ...
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1answer
224 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
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2answers
130 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
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1answer
148 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 ...
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1answer
261 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
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0answers
315 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 ...
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0answers
74 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 ...
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190 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 ...
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1answer
225 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.
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101 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 ...
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1answer
229 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
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2answers
202 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 $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
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3answers
575 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
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2answers
736 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 ...
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0answers
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
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1answer
328 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
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1answer
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 ...
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0answers
57 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 ...
4
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1answer
339 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
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0answers
91 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
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1answer
135 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
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1answer
291 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 ...
4
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1answer
82 views

The Chern connection on a Hermitian symmetric domain

There's a connection (the Chern connection) on the Tangent Bundle of a Kahler Manifold which is compatible with both the hermitan metric, and the holomorphic structure. In general, I guess there's no ...
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0answers
120 views

How to prove this inequality of heat flow from Weitzenbock formula?

Let $(M,g), (N,h)$be a compact Riemannian manifolds, $m:=\dim M, n:=\dim N\geq 2$, and $N$ is a non-positive curvature $K_N\leq 0$. All connections which appear below are the Levi-Civita connections. ...
6
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1answer
239 views

Lifting a Diffeomorphism to the Cotangent Bundle

Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving. ...
2
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1answer
159 views

How to find Darboux coordinates?

I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be ...
20
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1answer
529 views

Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...
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1answer
76 views

Non Linear PDE's [closed]

I want to solve two systems of questions being $$\frac{C(r,y)''}{C(r,y)} + \frac{C(r,y)'^2}{C(r,y)^2}=0$$ and $$A(r,y)'' + 2A(r,y)' \frac{C(r,y)'}{C(r,y)}=0$$ where ' is differentiating with respect ...
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0answers
69 views

The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$

Let $\Sigma_n,n\ge 1$ be a sequence of embedded minimal disks in $\mathbb{R}^3$ such that: (1) $0\in\Sigma_n\subset B(0,r_n)$ with $r_n\to\infty$ as $n$ tend to $\infty$, (2) ...
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1answer
195 views

The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
6
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0answers
128 views

The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...
3
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1answer
163 views

Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$. A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...
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1answer
71 views

Calculate GPS coordinates at x meters [closed]

I want to calculate a pair of GPS coordinates(lat,long) that is at x meters N/S/E/W from a known point (lat_old,long_old). I have found the Haversine formula ...
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0answers
69 views

Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...
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0answers
87 views

Computing the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
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1answer
192 views

What is the difference between $\delta W^{\pm}=0$ and Einstein?

Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference ...