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

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32 views

Bochner-Weitzenbock formula for flat bundle Laplacian

Suppose $(M,g)$ is a compact Riemannian manifold and $(E, \nabla, \lambda, B)$ is the following data: 1) $E$ is a complex vector bundle over $M.$ 2) $\nabla$ is a flat connection. 3) $B$ is a ...
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62 views

lie derivative on complex manifold

Let $M$ be a complex manifold with complex structure $J$. Then for any smooth vector field $X$ on $M$ can be splited into its holomorphic and anti-holomorphic components $X=Z+\overline{Z}$. For any ...
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49 views

Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary). A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$. The geodesic flow can be seen as a ...
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0answers
101 views

Proof of Arnold-Liouville theorem in classical mechanics [on hold]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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1answer
85 views

Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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85 views

Derivative of a group action [migrated]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
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1answer
135 views

Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group. Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
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1answer
191 views

Convex polyhedron and its Gauß-curvature [on hold]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here. What I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. ...
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174 views

Non-university jobs suited for pure mathematician turned computational neuroscientists, with coding experience [closed]

I also asked this question on academia stack exchange, http://academia.stackexchange.com/questions/48057/type-of-non-university-research-jobs-suitable-for-a-mathematician-turned-comput but asking ...
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1answer
192 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 ...
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1answer
156 views

Automorphism group of a fiber bundle surjects onto diffeomorphism group?

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with ...
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36 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
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1answer
64 views

Gradient of distance function at cut points on Alexandrov spaces

Let $M$ be an $n$-dim Alexandrov space with curvature bounded below $sec \geqslant k$, possibly non-compact. We assume that $M$ has no boundary for simplicity. For a compact subset $K \subset M$, the ...
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0answers
94 views

Steepest descent path and Picard-Lefschetz theory

Assume that an ordinary integral of the form $$I=\int_{-\infty}^{\infty}dx e^{-f(x)} $$ for some real function $f(x)$ is given where $f(x)$ is well defined over all $\mathbb{R}$ and the integral is ...
2
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0answers
97 views

Gauge freedom in the tetrad

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics": http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958 by Kinnersley. I have a ...
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0answers
156 views

How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence $\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...
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47 views

Characterisation of vector fields solution to a simple equation

This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ...
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2k 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, ...
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143 views

Smooth morse theory of Riemannian distance functions

Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...
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1answer
94 views

Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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0answers
90 views

Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...
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1answer
314 views

Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$ M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0. $$ My question is to ...
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2answers
250 views

Volume-minimizing submanifold implies calibrated?

Let $X$ be a smooth manifold of dimension $d$ and $M$ an oriented submanifold of dimension $p < d$ so that the multiples k⋅M are absolutely minimizing $p$-volume in their integral homology ...
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1answer
131 views

Marcel Berger's “Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes.”

I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.
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59 views

Bounding distance between geodesics in manifolds with nonpositive curvature

This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...
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0answers
21 views

Symplectic group action [migrated]

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
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0answers
59 views

Volume element of symmetric definite matrices in polar coordinates

I have a difficulty to understand the following statement. I don't ask for a proof but just understand the statement concretely (what it does mean, how to apply it...) Let $\mathcal P_n$ be the ...
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0answers
101 views

Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com, ...
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1answer
166 views

Positive solutions to Yamabe problem?

Yamabe problem ensures that for any Riemannian metric $g$, in its conformal class $[g]$ there always exists a metric $\bar g$ whose scalar curvature $\bar R$ is constant. I was wondering whether ...
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1answer
153 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
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1answer
149 views

Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
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274 views

Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For ...
2
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1answer
175 views

existence of totally geodesic hypersurfaces

Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...
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0answers
54 views

Kerr metric affine parameter

I'm going through the chapter about Kerr space-time of Chandrasekhar's "Mathematical theory of black holes", and have a question about the following transformation: the idea is, that one wants to ...
3
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0answers
57 views

Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer: Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...
4
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2answers
168 views

Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology. The evaluation map $$ev\colon ...
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0answers
112 views

Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...
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0answers
94 views

Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here: Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
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0answers
83 views

derivative of the adiabatic limit of the eta invariant

To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...
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0answers
67 views

Curvature tensor for a singular manifold

Given a manifold $M$ with its tangent space $TM$ and frame vector field $e \in TM$. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of $e$ with ...
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0answers
89 views

Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold

Let $(M,\omega)$ be a Kahler manifold with Kahler integral two-form $\omega$ and let $(L,h)$ be a rank-one complex vector bundle over $M$ equipped with a fixed hermitian metric $h$. I am interested in ...
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0answers
262 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 ...
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1answer
109 views

Extension of a smooth function from a convex set

Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear ...
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0answers
89 views

Incorrect statement in a proof of the transversality theorem? [migrated]

I'm reading through Morris Hirsch's book on differential topology, and he makes the following offhand statement. Suppose k is a compact subset of a manifold U, and V is a vector subspace of R^n. If a ...
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0answers
65 views

Differential form heat kernel on hyperbolic space

Is there an explicit formula in the literature for the heat kernel of the Hodge Laplacian on differential forms? I found some on functions, but not on forms of higher degree. What at least about ...
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0answers
20 views

Finding incomplete geodesics [migrated]

I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic? In particular, I am trying to solve the following exercise: Consider the upper ...
2
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2answers
416 views

Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...
5
votes
2answers
121 views

Differentiability of polytope shadow areas

Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$, and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$. For a point $x$ on $S$, let $\sigma(x)$ ...
3
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1answer
97 views

Generalisation of “tangent space” to not-necessarily connected sets

I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following. Definition (Blob) Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The ...
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19 views

Restrictions of potential tensor fields to toric subgroups

Let $G$ be a compact connected nonabelian Lie group and let $f$ be a symmetric tensor field of order $m\geq1$ on $G$. Let $T\subset G$ be a translate of a torus subgroup of $G$ with $\dim(T)\geq1$. ...