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

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7 views

Gauss Curvature Equation for hypersurface in Semi-Riemannian manifold

We have the Gauss curvature equation: $$\langle R(V,W)X,Y\rangle = \langle R'(V,W)X,Y\rangle - \langle II(V,X),II(W,Y)\rangle + \langle II(V,Y),II(W,X)\rangle$$ Here $M$ is an immersion in $N$. ...
-1
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0answers
37 views

Geometry of a submanifold of $\Bbb S^n$ [on hold]

Let assume that $S$ is a subspace of $\Bbb R^{n+1}$ and $M:=S\cap {\Bbb S^n}$, we know that $M$ is a submanifold of $\Bbb S^n$. I need to know the tangent space of $M$ at every $x\in M$ as well as the ...
0
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0answers
25 views

Homogeneous regular manifolds

In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way: ...
-2
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0answers
80 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$ [migrated]

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
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1answer
91 views

Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification. Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...
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0answers
31 views

Mathematical Billiards: Set of starting points and velocities hitting boundary at time t

In the simplest setting $\Omega$ smooth compact and convex in $R^n$ with linear constant speed trajectories that is ($q_t=q_0+t\cdot v$ until the collision point). What is known about the structure ...
1
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0answers
36 views

How to calculate Chern class for reflexive sheaf?

Let $(X,\omega)$ be a Kahler manifold of dimension $n$ and $\mathcal{F}$ a reflexive sheaf on $X$. Since there is no global resolution of sheaf by vector bundles in the non-projective manifolds. It ...
2
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0answers
51 views

Conditions for existence of Penrose diagrams

A Penrose diagram (also known as a conformal diagram or Carter-Penrose diagram) is a technique for visualizing the causal (light-cone) structure of a 3+1-dimensional manifold. Usually the diagram is ...
23
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219 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
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0answers
87 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...
2
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1answer
152 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
1
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1answer
64 views

The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the ...
4
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1answer
138 views

Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...
2
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1answer
88 views

Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
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0answers
114 views

Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It is clear that ...
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1answer
118 views

Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that $$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
1
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0answers
48 views

Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
3
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0answers
55 views

Perturbations to a vector field

I ran into some problems while working through a proof of the Poincare-Hopf theorem that essentially boiled down to the following question: given a smooth vector field $V$ on a (compact Riemannian) ...
3
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0answers
109 views

Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
12
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0answers
206 views

Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). ...
3
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1answer
95 views

Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...
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9answers
1k views

Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help. In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
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0answers
57 views

A compact Alexandrov space with curvature bounded below has curvature bouneded above? [on hold]

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
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91 views

Curvature in geometry-interpretation

Previously this question was asked on stack exchange: the answer contained only reference to the wikipedia page which I already read (as mentioned in my post). So here is the question: The are ...
7
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2answers
270 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
3
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0answers
71 views

Representing rational homotopy class by geometric objects

Given a smooth manifold $M$, we can study its rational homotopy type by looking at differential forms. I am wondering if there is a way to represent each $rational$ homotopy class by geometric ...
1
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0answers
62 views

Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
5
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0answers
130 views

Detecting torsion-classified bundles by differential invariants

The following is based on a loose understanding of the nuts and bolts that go into Chern-Simons theory, so bear with any vagueness on my part. Suppose I have a principal $G$-bundle $P\to M$ and I ...
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2answers
107 views

Does an analytic tensorial Lie structure on $S^2$ gives a fiberwise Abelian Lie algebra structure?

Motivated by the answer to this question we ask: Is it true to say that for every real analytic tensorial Lie algebra structure $\alpha$ on $\chi^{\infty}(S^2)$, all fibers are necessarily ...
3
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0answers
85 views

Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let $(M_i,p_i)$ be a sequence of $n$-dimensional Riemannian manifolds with lower Ricci curvature bound $-1$. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there ...
1
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1answer
116 views

Manifold_Lie algebra compatibility

In this question we try to improve some parts of this post as follows: What is an example of a manifold $M$ and a Lie algebra $L$ (with the same dimension) such that $M$ does not admit ...
5
votes
2answers
451 views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
5
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2answers
301 views

Riemannian metrics preserved by diffeomorphisms

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$? Note that ...
4
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3answers
190 views

Parameterizing rotations of a cube [closed]

For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...
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0answers
96 views

When is a conformal class equal to a conformal orbit?

Let $(M,g)$ be a Riemannian manifold of dimension $n$. Let $\text{conf}(M,g)$ denote the conformal group, i.e. the subgroup of diffeomorphisms of $M$ that acts by conformal transformations relative to ...
6
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1answer
213 views

Differential geometry without the Hausdorff condition or the second axiom of countability

I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things ...
6
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1answer
182 views

How does one identify flow lines on a vector bundle with those on the base in Morse theory?

In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function ...
4
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2answers
251 views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
5
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1answer
293 views

Is there a geometric proof for the upper semicontinuity of fiber dimension in algebraic geometry?

One of the first theorems encountered in algebraic geometry is the upper semicontinuity of fiber dimension: Let $ f : X \to Y $ be a surjective regular map between irreducible varieties with ...
14
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1answer
679 views

Is it possible for a metric on a smooth manifold to be smooth?

Are there any smooth manifolds $M$ with the following property: There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$? If not, is it ...
1
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0answers
119 views

Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
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1answer
66 views

Are non-linear connections with linear holonomy, linear?

Let $\pi\colon TM\to M$ be the tangent bundle of a differentiable manifold, let $E=TM\backslash 0$ be the slit tangent bundle, and let $V_eE$ be the kernel of $\pi_*$ at $e\in E$. The set ...
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0answers
73 views

Integral curves on non compact manifolds [closed]

Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in ...
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0answers
32 views

Normal fields of geodesic spheres

This question is related to this one (http://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...
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1answer
121 views

Volume form on pair (X,D)

Let $X$ be a singular Kahler variety with Kahler current $\omega $ then the volume form is $\omega^n$. Now let $D$ be a divisor then how can we define volume form on pair $(X,D) $?
3
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1answer
193 views

Generalising the parametric transversality theorem to a foliation

The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all ...
1
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1answer
146 views

Orientability of Surfaces and the Fundamental Group [closed]

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
3
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1answer
159 views

Orbits of Metrics under the Action of the Diffeomorphism Group

Consider the $n$-sphere $$ S^n = \{x\in\mathbb{R}^{n+1}: 1 - \sum_{k=1}^{n+1} x_k^2 = 0\}, $$ and let $g_1$ be the induced metric. Given $\lambda\in\mathbb{R}^{n+1}_{>0}$, we have the ellipsoid $$ ...
4
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0answers
156 views

Open questions in “Spin geometry”

This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics ...
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0answers
111 views

How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$ with cone singularities along divisor $D$ with cone angle $2\pi\beta$ is said to be of constant scalar curvature Kahler or cscK if its scalar curvature $S(\omega)$, which is ...