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

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1
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0answers
80 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
3
votes
1answer
111 views

When is the semidirect product of principal fiber bundles a fiber bundle

Let $P_{H}$ be a principal bundle over a manifold $M$ with fiber the Lie group $H$ and let $P_{G}$ be a principal bundle with fiber the Lie group $G$ over the same manifold $M$. Let $h_{ab}\colon ...
1
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0answers
109 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
2
votes
1answer
68 views

Reference for Bonnet Fundamental theorem of surfaces in Lorentzian spaces

I am looking for a reference for the following folklore theorem, which is the Lorentzian analogue of the Bonnet fundamental theorem of surface in Euclidean space, hyperbolic space or $3$d sphere. ...
1
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0answers
76 views

Pushforward for differentiable stacks/ Lie groupoids

Let $X$ be a differentiable stacks, and let $(G_{0}, G_{1}, s,t)$ be a Lie groupoid representing $X$. Let $NG_{\bullet}$ be the nerve of the above groupoid. The De rham complex of $X$ can be defined ...
2
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0answers
102 views

Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...
1
vote
1answer
74 views

Triviality of a circle fibration induced by an almost complex structure

Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More ...
4
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0answers
226 views

Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds) And Ricci curvature of Narasimhan-Simha ...
3
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0answers
111 views

Distance to the level sets of an almost linear function

While reading about the almost-splitting theorem of Cheeger and Colding, I thought about the following question, which asks about wether the level sets of a function which is close to being a linear ...
2
votes
0answers
121 views

The minimum value of a energy integral

Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0 $ and $${\nabla ...
0
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0answers
43 views

Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces. Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...
9
votes
2answers
478 views

Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique). I am ...
-1
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1answer
106 views

Parallel transport on simplicial manifold? [closed]

Do you know some reference about the notion of parallel transport for simplicial manifolds?
2
votes
0answers
60 views

Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...
2
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0answers
35 views

Largest disk inside a spherical domain

It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any ...
1
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0answers
48 views

Momentum Map on cotangentbundle as submersion

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
3
votes
0answers
68 views

How to visualize the dual objects of jets of functions?

I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...
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1answer
248 views

Well-known name for a certain connection

Have $X \subset \mathbb{R}^3$ be a smoothly embedded surface. Then we try to define a connection on the tangent bundle $TX$ as follows. The tangent space $T\mathbb{R}^3$ is naturally a trivial ...
4
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0answers
134 views

Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
1
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1answer
116 views

Applying Cheeger and Colding segment inequality

The question turns out quite long and maybe a bit vague, I apologize in advance for that. I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently ...
3
votes
0answers
43 views

Prescribed curvature problem of a connection beyond the real analytic category for $SL(3,R)$ bundles?

With reference to the questions Does the Riemann-Christoffel curvature determine the connection? and When is a given matrix of two forms a curvature form?, and recalling the following important result ...
2
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1answer
161 views

Tangent bundle of a homogeneous space and the euler exact sequence

Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$: $$0 \to \mathfrak{h} \to \mathfrak{g} \to ...
8
votes
1answer
135 views

Convex body with affine-equivalent cross-sections

I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that ...
2
votes
0answers
115 views

does there exist a generalization of a manifold [closed]

Does there exist a generalization of a manifold whereby instead of being locally $\mathbb{R}^n$, it's locally another specified space?
4
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1answer
289 views

Unusual inequality concerning elementary symmetric functions

To state the question and fix conventions I will introduce some notation from e.g. (Lin-Trudinger, Bull. Aust. Math. Soc. 1994, ``On some inequalities for elementary symmetric functions") Given ...
6
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2answers
122 views

Isolate flat umbilic on compact Riemannian surface with nonpositive curvature

We know that we can have a noncompact surface with a so called Monkey saddle (https://en.wikipedia.org/wiki/Monkey_saddle), on which there is a isolated flat umbilic (where the Gaussian curvature ...
1
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0answers
58 views

Possible directions of saddle connections

Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...
0
votes
0answers
40 views

trace sobolecv inequality for $q=2(n-1)/(n-2)-\varepsilon$ in half space

Can I do the following inequality, for $ u\in D^{1,2}(R^n_+)$, we have $(\int _{R^{n-1}} |u|^{2(n-1)/(n-2)-\varepsilon}dx')^{\frac{ 1}{2(n-1)/(n-2)-\varepsilon } }\leq C ( \int _{ R^n_+}| \nabla ...
9
votes
1answer
337 views

What are the different ways of defining 3-manifolds?

I wonder what are the different ways of defining the 3-manifold. Obviously for average human being it is difficult to imagine the 3-manifold. Therefore the presentation or visualisation of such object ...
28
votes
0answers
478 views

Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order to really understand this result rather broad background is required: you need to know analysis (pseudodifferential ...
1
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0answers
53 views

Dimension of tangent space to manifold of cross section slices

Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...
5
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0answers
233 views

On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$). If we suppose the curvature tensor $R$ of $g$ ...
1
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0answers
84 views

“Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...
10
votes
3answers
520 views

What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
4
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0answers
47 views

Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a ...
1
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1answer
55 views

Existence of left-invariant metric on the cotangentbundle of homogeneous spaces?

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
0
votes
0answers
51 views

Trace Theorem for $q< 2(n-1)/(n-2)$

Can I get a trace theorem inequalite for $R^n_+$: For $q\in [2,2(n-1)/(n-2]$, we have $(\int_ {R^{n-1} } |u|^q dx) ^{2/q}\leq C(\int_{R^n_+} |\nabla u |^2dx)^{1/2}.$
1
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0answers
60 views

Scattering in (pseudo-)Riemannian spaces

I will ask my question in a broad way, leaving a lot of freedom for answers. Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
5
votes
1answer
196 views

Local differential geometry and invariant theory

Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists? Start with ...
6
votes
2answers
154 views

Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5

Up to topology, the 5D homogeneous space $$ G_2/P $$ of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space $$ M:=J^1(2,1)=\{(x,y,u,p,q)\} $$ of scalar functions in two ...
6
votes
0answers
157 views

“Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations: $f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact. $f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...
0
votes
0answers
48 views

map of constant rank

Let $f_1, \dots, f_m \colon M^n \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m,n $ on $M$. I'm trying ...
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1answer
533 views

Is there a geometric construction of hyperbolic Kac-Moody groups?

Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...
3
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0answers
52 views

Interpolating from a Hard Lefschetz class to a Kaehler class

Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures. There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...
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0answers
63 views

Curves in $\mathfrak{su}(n)$ with specific property

Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...
1
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1answer
178 views

Can a conformal map be turned into an isometry? [closed]

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, ...
2
votes
1answer
93 views

Modification of Morse lemma with two functions

Standard Morse lemma states that if the singularity of a smooth function $f$ is non-degenerate, one can choose coordinates such that function has a "quadratic" form $f = \sum \limits_i x_i^2 - \sum ...
1
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1answer
121 views

Is a non-flat hermitian connection determined uniquely by its holonomy and curvature?

How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where $$\alpha\in\text{Hom}(\pi_1(M),U(1)),~~~~F\in ...
1
vote
1answer
165 views

Tangent complex of dgla/ twisted dgla

I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism. Let $(\mathfrak{g}, d, ...
2
votes
0answers
117 views

Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$? Think of $\Bbb R ...