**0**

votes

**0**answers

168 views

### When an Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When an Spherical variety is $K$-stable? Is ...

**11**

votes

**1**answer

341 views

### Schemes over topological rings

I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...

**1**

vote

**2**answers

152 views

### Defining Gauss-Kronecker curvature for submanifolds of $\Bbb R^n$ and relation with ${\rm d}{\bf N}_i$'s

I'm trying to find a definition for Gauss-Kronecker curvature of submanifolds of $\Bbb R^n$, but I'm only finding it for hypersurfaces. I would like to know if someone knows any text which works in ...

**1**

vote

**0**answers

113 views

### Hermitian metric on conic Kaehler-Einstein setting

I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler ...

**0**

votes

**1**answer

120 views

### Gluing submanifolds along their common boundary

This might be too elementary for this site, but I asked first on math.stackexchange and didn't get an answer even after offering 250 bounty points.
Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ ...

**2**

votes

**3**answers

333 views

### Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...

**2**

votes

**1**answer

239 views

### Gauss-Bonnet invariant Ω: explicit intrinsic expression for Π in Ω=dΠ?

Let the Gauss-Bonnet form be $\Omega\propto\text{Pf}(\Omega^i{}_j)$ with $\Omega^i{}_j$ the curvature 2-form of an even-dimensional manifold with dim=$n$. The Gauss-Bonnet form is exact, as shown in ...

**4**

votes

**0**answers

187 views

### Characterization of kernel of Bianchi operator

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the Frechet space of symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the Frechet manifold of metrics on ...

**0**

votes

**1**answer

127 views

### Ambient isotopy of the diagonal submanifold in product space

Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
...

**0**

votes

**0**answers

69 views

### Positive curvature of the boundary away form a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...

**3**

votes

**2**answers

317 views

### SU(2) and differential forms [closed]

I am a physicist with some background in differential geometry and I apologize for any possible unprecise terminology.
Consider the Lie group $SU(2)$ and its tangent space $su(2)$ forming a tangent ...

**9**

votes

**1**answer

314 views

### Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...

**2**

votes

**1**answer

204 views

### When is a homogeneous space connected? [closed]

Let $G$ be a Lie group (not necessarily connected) and let $H$ be a closed subgroup of $G$. I am after an algebraic (group theoretic) characterization of when the homogeneous space $G/H$ is connected.
...

**0**

votes

**0**answers

47 views

### Shuffle multiplication and generalized Leibniz rule in tensor calculus

The headline already says it: Is anybody (except me) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the commutative shuffle product ...

**5**

votes

**1**answer

341 views

### How many geometric structures on manifolds are there?

Let $M$ be a smooth manifold, of dimension $n$. I know that there are many types of geometric structures on $M$. A large number of them are captured by the notion of a $G$-structure, which is a ...

**3**

votes

**2**answers

254 views

### Fano manifold admit an smooth anti-canonical divisor?

Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$

**2**

votes

**0**answers

67 views

### The level set of convolution

Suppose $u\in C^2(\bar \Omega)\cap C^\infty(\Omega)$ is a function which is compact supported on $\Omega$ with $u\equiv 0$ at $\partial\Omega$, and $\Omega\subset \mathbb R^3$.
We also assume that ...

**6**

votes

**0**answers

212 views

### Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...

**3**

votes

**0**answers

87 views

### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...

**4**

votes

**0**answers

217 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**4**

votes

**1**answer

226 views

### Besse p134 Riemann tensor in dimension 4

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...

**5**

votes

**2**answers

226 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 $f^{-1}(f(x))$.
...

**1**

vote

**0**answers

41 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 ...

**1**

vote

**1**answer

130 views

### Sequence of smooth maps converging to the identity [closed]

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 ...

**1**

vote

**1**answer

63 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

78 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 ...

**2**

votes

**0**answers

62 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

102 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

68 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 ...

**13**

votes

**7**answers

757 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, ...

**4**

votes

**1**answer

199 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 ...

**-1**

votes

**1**answer

114 views

### cartan killing metric [closed]

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

281 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 ...

**7**

votes

**0**answers

170 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 ...

**1**

vote

**2**answers

138 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

**2**answers

265 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 ...

**1**

vote

**1**answer

123 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 ...

**0**

votes

**1**answer

154 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 ...

**3**

votes

**1**answer

84 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, ...

**1**

vote

**1**answer

108 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 ...

**0**

votes

**2**answers

339 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 ...

**4**

votes

**1**answer

431 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 ...

**1**

vote

**1**answer

177 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
$$|p_t(x,y)| \leq C$$
...

**1**

vote

**1**answer

346 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 ...

**10**

votes

**2**answers

498 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 ...

**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 ...

**6**

votes

**1**answer

255 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 ...

**2**

votes

**0**answers

142 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

93 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 ...

**5**

votes

**1**answer

139 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 ...