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

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1answer
92 views

Analogue of fundamental theorem of real surfaces for complex surfaces

Is there an analogue of the fundamental theorem of surfaces for complex surfaces? If I know only differentiable functions $E,F,G,e,f,g$ (coefficients of the first and second fundamental forms) where ...
2
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1answer
48 views

First Chern class and second Chern class in Quantizable Kaehler manifolds

Assume that $(X,\omega)$ is a K\"ahler manifold and $L\to X$ be a pre-quantum line bundle, then is there any relation between first Chern class and second chern class?
5
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2answers
86 views

References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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0answers
61 views

How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields. I dont know ...
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0answers
315 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...
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2answers
241 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
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0answers
39 views

Submersion on open and dense subset

If $f \colon M \to N$ is a smooth map between smooth manifolds, is it possible to find an open and dense set $M_0$, such that $f(M_0)$ is a manifold and $f \colon M_0 \to f(M_0)$ is a surjective ...
6
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1answer
364 views

what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$. What is G'? I know there is concrete description in terms of pairs ...
9
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2answers
154 views

Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
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40 views

Lower bounds on the measure of balls in attractor sets

I'm looking for a source for the following result. Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset ...
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2answers
134 views

Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies ...
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39 views

Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform. I know that if we perform a tetrad rotation, say of Class III: $(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, ...
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140 views

When are Kähler potentials bounded from below?

The prototypical example of global Kähler potential is the one of the standard Kähler structure on $\Bbb C^n$ given by $$f:\Bbb C^n\longrightarrow \Bbb R,\quad f(z_1,\ldots,z_n)=\sum_{k=1}^n|z_k|^2.$$ ...
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138 views
+50

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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56 views

Lower bound on the diameter of a ball contained in the stable manifold of a critical point

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Consider the negative of the gradient ...
6
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1answer
227 views

“structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration". Does it make sense to talk about "structure ...
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0answers
123 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie ...
9
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1answer
275 views

A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning ...
2
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1answer
77 views

Volume of the subelliptic ball

Let $\Omega \in \mathbb{R}^n$ a bounded open set when $n\geq 2$, and let $X_{1},X_{2},\cdots,X_{m}$ be real smooth vector fields that satisfy Hormander condition on $\Omega$. If we denote $Q(x)$ as ...
2
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1answer
162 views

Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is ...
3
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93 views

Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...
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0answers
155 views

Exact sequence of vector bundles

Consider the short exact sequences below; \begin{equation} 0\longrightarrow H^0(\mathbb{P}^4,\mathcal{O}_{\mathbb{P}^4}(d-1)^{\oplus 4})\longrightarrow ...
3
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0answers
30 views

covariant derivative of manifold-valued function and logarithm map

Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon ...
2
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1answer
169 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
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0answers
38 views

Feynman-Kac formula and time-ordering for vector bundles

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ ...
1
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1answer
37 views

Hermitic connections on complex line bundles with imaginary curvature form

It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
2
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0answers
94 views

Quantizable vs. integral Kahler form

Let $(M,\omega)$ be a (not necessarily compact) Kahler manifold. Then the form $\omega$ is integral if and only if $\omega \in c_1 (L) $ for some holomorphic line bundle $L$. A Hermitian holomorphic ...
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0answers
54 views

Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
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4answers
311 views

Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
3
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1answer
134 views

Compact manifolds locally bi-Lipschitz to Euclidean space

I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
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0answers
146 views

Is there any notion of “smoothification” from $\mathbb{R}$-schemes to generalized smooth spaces?

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor ...
8
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0answers
95 views

Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...
5
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0answers
76 views

Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...
2
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0answers
252 views

Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
2
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0answers
279 views

Do Peano curves provide a counterargument to Grothendieck's critique?

This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...
29
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1answer
850 views

Can a topological manifold have different tangent bundles?

We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of ...
1
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0answers
21 views

Normal cycle of union of convex sets [closed]

i have to show that the normal cycle of a convex subset is additive, i.e if A and B are two convex subsets then N(AUB)=N(A)+N(B)-N(A \inter B), i tought about using the Gauss-Bonnet formula since we ...
1
vote
1answer
49 views

Rectifiable currents [closed]

I found so many definitions of a rectifiable current, which is obviously a current which arises from rectifiable sets, but i really can't get the geometrical meaning of it. I saw some examples of ...
2
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0answers
36 views

Boundary Conditions for Sine Gordon on some K< 0 surfaces

The Sine-Gordon equation $$ \alpha^{\prime \prime}(s) = \sin \alpha (s) $$ defines asymptotic lines of all constant negatively Gauss curvature K surfaces on a Chebychev net in 3-space or so I ...
2
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0answers
57 views

What is known for harmonic map flow in dimension > 2?

I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. ...
3
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1answer
204 views

Confusion surrounding the Koszul-Malgrange theorem

I recently had the need to appeal to some complex geometry in my research and have been trying to unravel the various relationships surrounding the Koszul-Malgrange theorem. According to nlab, the ...
4
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0answers
71 views
+50

Compressing a hypersurface on the sphere

Let $M^n$ be a compact, connected, orientable hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$. Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero ...
3
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1answer
172 views

Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
5
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1answer
98 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is ...
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1answer
131 views
4
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1answer
89 views

Chern-Einstein metrics on complex Hermitian manifolds

Metric on a Riemannian manifold $(M,g)$ is Einstein, if for some function $\lambda\colon M\to \mathbb R$ $$ Ric(g)=\lambda g. $$ It is well know, that such $\lambda$ is, in fact, a constant. The ...
1
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1answer
47 views

Unbounded convex domains in 2D

Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior ...
5
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0answers
171 views

Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
2
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1answer
98 views

Hessians on Kahler Manifolds

This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, ...
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134 views

A question about invariance of plurigenera

Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the ...