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

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84 views

Is holomorphic 2-form on Moduli of Higgs bundle in Biswas' paper is non-degenerate

In Biswas' paper, Geometry of moduli of Higgs bundles, he defined a holomorphic 2-form on moduli of stable Higgs bundles, using Kodaira-Spencer map and Petersson-Weil metric. I want to know whether ...
1
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0answers
128 views

Is there any progress on Problem 13 (from Schoen and Yau)?

This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ ...
1
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2answers
142 views

Calculating Exterior Distance from Measurements of Inner Geometry

Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...
0
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0answers
90 views

Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
3
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1answer
128 views

Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle. I ...
2
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0answers
250 views

A metric on $S^{2}$ [on hold]

Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel z\parallel^{2}-\parallel w\parallel^{2},\;2z\bar{w})$. Define a metric on $S^{2}$ as follows: $$d(x,y)=Hd(p^{-1}(x), ...
10
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3answers
347 views

Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

I apologize in advance if this question is too vague for mathoverflow. My main aim is to get some references for a concept. First, we make the following observation: let $X: M \rightarrow TM $ be a ...
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0answers
45 views

Convergence of Selberg type Zeta function

Let $X=G/K$ be a Riemannian symmetric space without compact factor. We can assume $G$ is connected and real reductive. $K$ is a maximal subgroup of $G$. Let $\Gamma$ be a discrete torsion free ...
3
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1answer
235 views

Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic?

I'm learning differential geometry. I'm curious that when we learned analysis, we learned higher order derivative, while in differential geometry, first order derivative is generalized to element of ...
2
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0answers
116 views

Generalized metric on spacetimes

I read many articles about space-times. Most authors consider these spaces as warped product manifolds $I\times M$ where $I$ is an open connected interval of the real line and $M$ is a Riemannian ...
2
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0answers
86 views

Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?

To be more precise I am interested in questions similar to the one below (I asked the question below on math.stackexchange last week but got not answer.) I have a $C^1$ function $f:[0,1]^2 \to ...
2
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0answers
63 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
7
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2answers
883 views

Is there an English translation of Minding's 1839 paper?

Is there an English translation of "Wie sich entscheiden lässt, ob zwei gegebene krumme Flächen auf einander abwickelbar sind oder nicht..." by Ferdinand Minding, Journal für die reine und angewandte ...
3
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0answers
102 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
3
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0answers
85 views

An example of mean curvature flow that does not preserve embeddedness

Let $F: M^n \to \mathbb R^{n+k}$ be an embedding and $F_t$ be a families of immersions so that $F_0=F$ and $$\frac{\partial F_t}{\partial t} = \vec H$$ It is known that in hypersurface case ...
0
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0answers
87 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and ...
10
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3answers
712 views

Manifolds admitting flat connections

For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
2
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2answers
252 views

Monge-Ampere type PDE

NB: I have edited this question to clarify what the OP is asking – Robert Bryant Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = ...
4
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2answers
141 views

Reference for when a metric on a four-manifold is Kahler?

In a paper of Derdzinski (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...
3
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0answers
98 views

Surfaces ruled through a subset of points

One common definition of a ruled surface $S$ is that, through every point $p \in S$, there passes a line $L(p)$ that lies in $S$: $L(p) \subset S$. My question is: Q0. Is there any loosening of ...
2
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2answers
134 views

Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...
3
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1answer
71 views

Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that $X^{\mathrm{hor}}$ is a ...
0
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1answer
134 views

Hilbert's Theorem relevance to positive curvature

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in $ R^3 $. This theorem answers the ...
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0answers
98 views

Topological restrictions from mean curvature bounds

Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere. What happens when the mean curvature is small, or bounded? (For ...
2
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0answers
106 views

Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a ...
2
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0answers
56 views

Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...
3
votes
5answers
260 views

Variation of curvature with respect to immersion?

Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by $$ f(t) = f_0 + tuN_0, $$ where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...
6
votes
1answer
373 views

Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
-3
votes
1answer
149 views

The logarith map as a contraction [closed]

Two Questions: (1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping? Or more ...
5
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1answer
261 views

Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...
13
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1answer
305 views

Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...
1
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0answers
102 views

The relation between Gromov hausdorff convergence and inverse limit of compact metric spaces [closed]

The relation between Gromov hausdorff convergence and inverse limit of compact metric spaces.
2
votes
1answer
231 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
3
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0answers
90 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
0
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0answers
131 views

A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions". Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...
11
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3answers
1k views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
4
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1answer
276 views

Smooth and $GL(n)$-equivariant implies algebraic?

Context: Let $B_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L_n\subset B_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite ...
5
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0answers
174 views

Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...
2
votes
1answer
318 views

The complex structure on $S^{2n}$

Assuming that there is a complex structure on $S^{2n}$ and it becomes a complex manifold, also assuming there are complex coordinate $z, w$ on $U, V$ respectively, where $U, V$ are open cover of ...
3
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0answers
275 views

How to estimate $|\nabla T|^2\geq c|\delta T|^2$ besides Cauchy-Schwarz inequality?

Let $T$ be a $(0,k)$-tensor on a Riemannian manifold. I was wondering how to improve the estimate $|\nabla T|^2\geq c|\delta T|^2$ for special metrics besides Cauchy-Schwarz inequality? The question ...
0
votes
2answers
173 views

Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...
2
votes
1answer
179 views

Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?

Let $G$ be a connected compact Lie group, consider the left/right action on itself. For any finite $A\subset G$, consider the centralizer $Z_G(A):=\{g\in G| a g= g a\}$. Q: is $Z_G(A)$ a connected ...
3
votes
1answer
196 views

Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete space of constant negative curvature, that is, a hyperbolic space, $\Gamma$ a discrete group of isometries, and and $M=H/\Gamma$ its quotient space; we assume ...
3
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1answer
88 views

Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...
2
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0answers
238 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
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0answers
82 views

Riemann curvature of $S^1$-principal bundle

Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have, $$T_pP \simeq T_pV\oplus\Gamma_p$$ Where $V$ ...
2
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0answers
102 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
1
vote
1answer
111 views

Affine differential geometry. Is Calabi's hypersurface isotropic?

I am in the framework of (equi)affine differential geometry. Let $E$ be a centro-equiaffine space, that is a real vector space of dimension $n$, together with the special linear group $SL_n(R)$. Let ...
20
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1answer
599 views

Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph: " ...In a certain sense, this might be considered a very satisfactory generalization of Guass-Bonnet. ...
7
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1answer
685 views

Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...