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

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1answer
51 views

Normals along a Sphere

Let $M \subset \mathbb{R}^d$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) ...
0
votes
1answer
163 views

Are compact complete geodesics closed? [on hold]

note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO. check this link: ...
19
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0answers
221 views

Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$. This is a not too difficult theorem due to Whitney, proved in many textbooks. ...
0
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0answers
23 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...
1
vote
1answer
289 views

Variation formula of a metric [on hold]

In Terry Tao's notes on the Poincare Conjecture, he makes a jump I can't understand. From differentiating the identity $g^{\alpha \beta}g_{\beta \gamma} = \delta^\alpha_\gamma$ we obtain the ...
0
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1answer
995 views

Categories of Geometry [closed]

I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified. It seems Euclidean Geometry, Affine ...
1
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0answers
58 views

Extending integrable almost-complex structure

Suppose $(X,I)$ is an almost-complex real analytic manifold where $I$ is a real-analytic almost complex structure. Suppose there exists an $I$-almost complex submanifold $M\subset X$ where this means ...
6
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1answer
200 views

Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds: Find a pair of complex analytic families $\lbrace M_t\rbrace$ and ...
7
votes
1answer
560 views

How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
1
vote
1answer
120 views

Normal Variation on Manifolds

Let $M$ be a smooth surface and let $x,y \in M$. Let $d_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum ...
0
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0answers
71 views

Integral of Weingarten Map / Shape Operator [on hold]

This Paper states that the Weingarten Map / The Shape operator $W_p$ of a two-dimensional surface $S\subset\mathbb{R}^3$ at a point $p$ can be expressed in the following way: ...
7
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0answers
129 views

Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...
4
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0answers
621 views

Question on Atiyah-Patodi-Singer on $T^3$

I wanted to test my understanding of the Atiyah-Patodi-Singer theorem by studying flat bundles on $T^3$ explicitly, and miserably failed. Namely, I computed the eta invariant explicitly for flat ...
4
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0answers
213 views

If 2-manifolds are homeomorphic and smooth, are they diffeomorphic? [on hold]

Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. ...
4
votes
1answer
286 views

When is a `1-form' with continuous coefficients exact?

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in ...
1
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2answers
245 views

Notion of manifold curvature?

Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is ...
3
votes
1answer
92 views

Transitivity of the action of the group of gauge transformations on the space of hermitian metrics

This is a cross-post from math.SE Let $E \to M$ be a complex vector bundle with $P$ the associated $GL(n,\mathbb C)$ frame bundle. The group of gauge transformations is the space of sections of ...
3
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2answers
150 views

Stability of minimal surfaces

Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
0
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0answers
194 views

Integral of Square of Mean Curvature

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus. When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...
5
votes
1answer
861 views

Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...
0
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0answers
32 views

An obstruction to existence of invariant trivializations of central extensions in the presence of a group action

This is sort of a follow up question to this. My real question is a particular case of the following abstract situation. Let $G$, $\pi$ be groups and assume $\pi$ acts on $G$ by automorphisms. Assume ...
4
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0answers
38 views

Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions. By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...
4
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4answers
272 views

Breaking up the free Lie algebra into Gl irreps

The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra , the free Lie algebra generated by any choice of ...
14
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5answers
532 views

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

First, we make the following observation: let $X: M \rightarrow TM $ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$ i_X \circ i_{X} ...
2
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0answers
68 views

Gromov width of cotangent disk bundle

Given a symplectic manifold $(M^{2n},\omega)$, the Gromov width of $M$ is defined to be $w(M)=sup\{{\pi r^2| B^{2n}(r) \rightarrow M}\}$ My question is: what is the explicit value of $w(D^*S^n)$, ...
0
votes
1answer
310 views

About the curvature of a connection? [closed]

In "Lectures on gauge theory and integrable systems" of M.Audin, she identifies the space of connections $\mathcal{A}$ on the trivial bundle $G\times S$ ($G$ Lie group, $S$ surface with boundary) with ...
0
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0answers
87 views

Quantization of Chern number $c_1^n$ on 2n dimensional spin manifold [closed]

All orientable 2-manifolds are spin manifolds, and we know that the quantization of the first Chern number $c_1$ of a complex line bundle on 2-manifold is $\mathbb{Z}$. For 4-manifolds, the second ...
6
votes
2answers
313 views

Is displacement controled by stable norm?

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a ...
4
votes
0answers
191 views

Isometries of hyper-Kähler manifolds

For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...
15
votes
1answer
457 views

Soft and hard part of geometry [duplicate]

While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" ...
3
votes
2answers
627 views

Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
1
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0answers
199 views

Invariance of a tensor Laplacian

Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let ...
25
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6answers
2k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
5
votes
2answers
492 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
1
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0answers
86 views

Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you. Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ ...
1
vote
1answer
125 views

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
-3
votes
0answers
61 views

Verification of Gauss Bonnet Theorem on Beltrami pseudosphere and bent sphere patches [closed]

Given that boundary geodesic curvature k_g and Gauss curvature K are constant, patch area = A and perimeter length = p. $ K\, A + k_g\, p = 2 \pi $ For a flat circle patch $ k_g= 1/R, $ $ ...
3
votes
1answer
113 views

Gaussian Curvature of Exponentiated 2-Planes

Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
5
votes
0answers
156 views

Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
0
votes
1answer
97 views

Green's function and eigenvalues with multiplicity

Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
8
votes
1answer
768 views

Central extension of the algebraic loop group

I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...
10
votes
1answer
768 views

Reference request for TQFT, functoriality

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds. It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says; The function $(M, \partial_{-}M, ...
-3
votes
1answer
379 views

Why does the Lefschetz Operator not Square to Zero? [closed]

I'm trying to learn about the Lefschetz decomposition but am having a very basic problem: For the fundamental form $K$ of a Kahler metric on a complex manifold $M$, the corresponding Lefschetz ...
0
votes
1answer
66 views

Is the kernel of a map between finite dimensional vector bundles still of finite type?

I'm not sure whether the level of this question is suitable for Mathoverflow. Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow ...
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0answers
110 views

Minimal surfaces are area-minimizing in small balls

A known fact about minimal surfaces is that they minimize the area functional on small balls. What proofs do you know of this? Update. Perhaps I should have been more explicit. My intention with this ...
8
votes
1answer
186 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 ...
7
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0answers
153 views

Reconciling the affine grassmannian and the based loop group

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...
5
votes
1answer
270 views

The properness of a submersion

Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is ...
9
votes
0answers
170 views

Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...
2
votes
0answers
119 views

Why does Hodge decomposition fail in the pseudo-Riemannian case?

Why does Hodge decomposition fail in the pseudo-Riemannian case? Does there exist a special class of pseudo-Riemannian manifolds for which it does not fail, for example Lie groups?