# Tagged Questions

**3**

votes

**0**answers

46 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**

votes

**2**answers

135 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 ?
...

**0**

votes

**0**answers

28 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 ...

**2**

votes

**2**answers

130 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 ...

**0**

votes

**0**answers

38 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, ...

**2**

votes

**0**answers

114 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
$$f:\Bbb C^n\longrightarrow \Bbb R,\quad f(z_1,\ldots,z_n)=\sum_{k=1}^n|z_k|^2.$$
...

**6**

votes

**0**answers

121 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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

217 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 ...

**6**

votes

**0**answers

120 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**

votes

**1**answer

267 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**

votes

**1**answer

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**

votes

**1**answer

161 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**

votes

**0**answers

92 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 ...

**1**

vote

**0**answers

151 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**

votes

**0**answers

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**0**answers

92 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 ...

**1**

vote

**0**answers

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 ...

**11**

votes

**4**answers

309 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**

votes

**1**answer

133 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 ...

**9**

votes

**0**answers

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**

votes

**0**answers

93 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**

votes

**0**answers

75 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**

votes

**0**answers

237 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**

votes

**0**answers

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**

votes

**1**answer

847 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**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

56 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**

votes

**1**answer

201 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 ...

**3**

votes

**0**answers

42 views

### 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**

votes

**1**answer

170 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**

votes

**1**answer

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 ...

**2**

votes

**1**answer

131 views

**4**

votes

**1**answer

87 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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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, ...

**0**

votes

**0**answers

133 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 ...

**4**

votes

**1**answer

117 views

### how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...

**1**

vote

**0**answers

46 views

### Is Fano Kahler surface with reverse orientation also Kahler?

In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?

**3**

votes

**0**answers

49 views

### Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it?
Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...

**1**

vote

**0**answers

72 views

### Flat coordinates of a Riemannian metric [closed]

We know that, given a metric $g_{ij}(p)$ on a $n$-dimensional manifold $M$, if the metric has vanishing Riemannian curvature, then there exists a coordinate system in which it is constant.
Let ...

**4**

votes

**0**answers

164 views

### Kähler quotients of affine varieties and GIT

Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...

**5**

votes

**1**answer

167 views

### Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...

**9**

votes

**1**answer

86 views

### Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...