**1**

vote

**0**answers

48 views

### Killing vector fields on sphere and nodal sets

Let $u$ be a smooth function on $S^2$ such that $\int_{S^2} ux_j =0$, for $j=1,2,3$. Does there exist a killing vector field $V$ on $S^2$ such that the nodal line of $\varphi=V(u)$ ($\{x\in S^2: ...

**4**

votes

**2**answers

148 views

### Geodesics and Riemannian submersions

Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion.
Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.
...

**12**

votes

**7**answers

609 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**11**

votes

**3**answers

2k views

### Commutator of Lie derivative and codifferential?

Let $(M,g)$ be some smooth, Riemannian manifold. Let $d$ be the exterior derivative and $\delta$ the codifferential on forms. For a smooth vector field $X$, let $L_X$ be the Lie derivative associated ...

**1**

vote

**1**answer

109 views

### Sequence of smooth maps converging to the identity

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

**3**

votes

**1**answer

222 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 ...

**1**

vote

**0**answers

28 views

### Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$

For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k ...

**0**

votes

**1**answer

47 views

### Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.

**1**

vote

**1**answer

72 views

### Semi-riemannian hypersurfaces

Let us consider the pseudo-euclidean space $\mathbb{R}^3_1$; that is, the space $\mathbb{R}^3$ endowed with the metric
$$\langle x, y\rangle = x_1y_1+x_2y_2-x_3y_3.$$
I see in O'Neill's book that ...

**1**

vote

**0**answers

46 views

### Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems ...

**0**

votes

**0**answers

82 views

### Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...

**0**

votes

**0**answers

53 views

### What does deg=0 imply for the gauss map?

I am facing the following problem. I have a Riemannian manifold $(M,g)$ with gauss curvature zero, an isometric immersion $v:M\rightarrow \mathbb{R}^3$ that is $C^{1,\alpha}$ and I consider the Gauss ...

**57**

votes

**11**answers

13k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**6**

votes

**2**answers

240 views

### Is every $S^3$ block bundle over $S^4$ a fiber bundle?

I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...

**14**

votes

**4**answers

1k views

### How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...

**4**

votes

**1**answer

160 views

### Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$.
Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...

**2**

votes

**1**answer

224 views

### Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...

**-1**

votes

**1**answer

91 views

### cartan killing metric [on hold]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**11**

votes

**1**answer

232 views

### Thom conjecture in CP3

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...

**1**

vote

**2**answers

128 views

### there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H

"Suppose that X is a compact projective manifold
equipped with a K¨ahler metric ω. Let L be a holomorphic line bundle
In general, there exists a hypersurface H ⊂ X such that
X \ H is Stein and L is ...

**6**

votes

**0**answers

150 views

### What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in ...

**5**

votes

**1**answer

948 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**

votes

**0**answers

76 views

### Generalized spin connection and dreibein in higher spin gravity

I am studying higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory.
It is well known ...

**5**

votes

**4**answers

835 views

### Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa?

wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I ...

**0**

votes

**1**answer

141 views

### existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...

**1**

vote

**1**answer

95 views

### Twistors for spaces of $n-$dimensions

I was wondering if there is any general definition for twistors for spaces of any dimension with a definite (or indefinite) metric; for example, in $\mathbb{R}^3$.
Twistors are spinors of the ...

**3**

votes

**1**answer

78 views

### Certain construction of the Itô integral on manifolds

Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, ...

**0**

votes

**0**answers

50 views

### Lift of isometries of spherical space forms [closed]

If we have an isometry between two spherical space forms, then it is said that it lifts to an isometry of the sphere. Why is that?

**1**

vote

**2**answers

374 views

### Different definitions of spin structures

This is the definition of spin structure according to Wikipedia:
which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one ...

**1**

vote

**0**answers

75 views

### Immersability and applications of a particular Riemannian metric

Let $N:\mathbb{R}^n\longrightarrow \mathbb{S}^{n-1}$ be a unit vector field and let $\alpha\in\mathbb{R}$ be constant. Does the Riemannian metric
$$G=\mbox{I}_n+\alpha N\otimes N$$
play any role in ...

**5**

votes

**2**answers

538 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 - ...

**0**

votes

**1**answer

131 views

### Chow stability and K-stability

Let $(M,L) $ be a polarized projective variety and is Chow stable, then under which condition it is K-stable in sense of Donaldson's definition?
Here is a good referrence for the definitions of Chow ...

**8**

votes

**3**answers

726 views

### A reference for smooth structures on R^n

There is a theorem stating that there is essentially one smooth structure on $R^n$ for every n other than 4. Does anybody know where i could find the proof of this? Not so much of what happens in ...

**4**

votes

**1**answer

406 views

### Elementary Proof of the Uniqueness of Smooth Structures on R

Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...

**5**

votes

**1**answer

125 views

### Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$

A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...

**9**

votes

**1**answer

855 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 ...

**1**

vote

**1**answer

159 views

### Heat kernel upper bound on compact Riemannian manifold

Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p_t(x,y)| \leq C$$
...

**0**

votes

**0**answers

135 views

### A converse to the Gauss Egregium theorem [closed]

Gauss Egregium theorem states that if there is full isometric mapping /bending ( when differential $ \Delta s $ length and $ \alpha $ angle between parametric lines are both conserved), then Gauss ...

**1**

vote

**1**answer

331 views

### Compact riemannian manifolds with boundary that have infinite volume?

I am looking for references in the literature pertaining to (essentially riemannian) metric spaces that are compact of infinite volume, such in the following example. Consider a riemannian metric on ...

**5**

votes

**1**answer

154 views

### Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere ...

**10**

votes

**2**answers

465 views

### Characterization of the exterior derivative

This is a cross-post of someone else's question.
I am cross-posting this question from MSE since it hasn't received any answers.
In the paper Natural ...

**6**

votes

**1**answer

245 views

### Which surfaces have only a finite number of connecting geodesics?

Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$,
under which conditions is it true that, for every pair of points
$a,b \in S$, there are an infinite number of ...

**4**

votes

**0**answers

85 views

### Continuity of the curve-shortening flow with respect to the curve

The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in ...

**2**

votes

**0**answers

135 views

### 3-manifold rigidity?

Defintion:
a $n$-manifold $M$ is said rigid if any homotopy equivalence $M\rightarrow N$ is homotopic to a homemorphism, where $N$ is an $n$-manifold.
The sphere $S^{3}$ and hyperbolic compact ...

**0**

votes

**0**answers

69 views

### Lagrangian fibrations with isolated singular fibers

Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold ...

**33**

votes

**3**answers

2k views

### Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...

**1**

vote

**1**answer

142 views

### Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...

**5**

votes

**0**answers

233 views

### Motivation behind Euler Theorem in differential geometry [migrated]

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...

**1**

vote

**1**answer

544 views

### Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as
...

**28**

votes

**6**answers

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 ...