**1**

vote

**1**answer

785 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
...

**0**

votes

**0**answers

8 views

### $d\bar d$-lemma on pair $(X,D)$

Let $X$ be a Kahler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kahler forms in the same Kahler class then have we $d\bar d$-lemma on pair ...

**0**

votes

**0**answers

23 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**3**

votes

**0**answers

45 views

### Plurisubharmonic functions on Kähler manifolds, intuition?

As the question suggests, what is the intuition for working with plurisubharmonic functions on Kähler manifolds?

**3**

votes

**1**answer

283 views

### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...

**0**

votes

**0**answers

80 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**4**

votes

**1**answer

123 views

### Minimal immersions of the 2-sphere

Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his ...

**0**

votes

**1**answer

149 views

### Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification.
Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...

**2**

votes

**1**answer

114 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...

**2**

votes

**1**answer

177 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**14**

votes

**4**answers

870 views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**5**

votes

**1**answer

470 views

### semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in ...

**3**

votes

**2**answers

324 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**2**

votes

**1**answer

123 views

### derivative of the adiabatic limit of the eta invariant

To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...

**4**

votes

**1**answer

96 views

### normal form of currents?

(this question did not get any answers on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space ...

**11**

votes

**1**answer

1k views

### Theorem of Bryant in higher dimensions

I have the following question. I read about Bryant's theorem which says that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically embedded as ...

**1**

vote

**0**answers

46 views

### Uniform bounds on affine invariants of a family of manifolds

Suppose $\{M_i\}$ is a family of $C^{\infty}$ smooth, strictly convex hypersurfaces in $\mathbb{R}^n$ such that $B_r\subseteq M_i\subseteq B_R$, where $B_r,B_R$ are balls centred at the origin of ...

**2**

votes

**1**answer

74 views

### Curvature computations of globally symmetric spaces of rank $1$

I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T_pM ...

**10**

votes

**1**answer

368 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**12**

votes

**1**answer

820 views

### Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...

**-4**

votes

**0**answers

146 views

### Almost complex structures on compact surfaces

Let $M$ be a real manifold of dimension $n$ and $E\rightarrow M$ be a rank two vector bundle above it. Let $J_0$ and $J_1$ be two almost complex structures on $E.$ If there is a continuous ...

**68**

votes

**11**answers

9k views

### Is it possible to capture a sphere in a knot?

You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...

**1**

vote

**0**answers

195 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**1**

vote

**0**answers

243 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**5**

votes

**0**answers

345 views

### When a 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 a Spherical variety is $K$-stable? Is ...

**0**

votes

**1**answer

234 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**1**

vote

**1**answer

74 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

**22**

votes

**3**answers

4k views

### What is the difference between holonomy and monodromy?

What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?

**5**

votes

**0**answers

138 views

### Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary

It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in ...

**4**

votes

**1**answer

166 views

### Zeta-Determinant Theorem

Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...

**9**

votes

**3**answers

698 views

### Is each closed convex set a manifold with corners?

Assume that $C$ is a convex set in $\mathbb{R}^{n}$ with non empty interior.
Then consider its closure, is it a smooth manifold with corners?
Edit:
1) The closure of $C$ should be a smooth manifold ...

**3**

votes

**0**answers

78 views

### about transverse complete intersection

There are several questions about transverse complete intersection arising from L. Guth's paper:
http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html
We say a polynomial ...

**3**

votes

**1**answer

60 views

### compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21:
Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...

**2**

votes

**0**answers

210 views

### Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$

Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for ...

**8**

votes

**2**answers

120 views

### Tangent fields spanning the distribution of principal directions on a surface

Suppose $S$ is an orientable regular surface in $\mathbb R^3$ without umbilical points (not necessarily compact, and with no boundary). There are two well-defined smooth $1$-dimensional tangent ...

**4**

votes

**0**answers

244 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**2**

votes

**0**answers

245 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**12**

votes

**2**answers

463 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**5**

votes

**1**answer

248 views

### Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...

**7**

votes

**1**answer

246 views

### Area of square to wrap a torus

The Nash-Kuiper
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
has recently been spectacularly visualized by the
Hevea Project.
This suggests two questions.
Q1. What is the area of ...

**4**

votes

**1**answer

344 views

### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

**2**

votes

**1**answer

148 views

### Are the Sasaki metrics on tangent and cotangent bundle isomorphic?

Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...

**2**

votes

**1**answer

386 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

**3**

votes

**0**answers

51 views

### If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?

In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem:
Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all ...

**1**

vote

**0**answers

59 views

### Singularities of quantum propagator in the case of piecewise constant controls

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{15} \rightarrow SU(4)$:
$V : (w_1, \ldots, w_{15}) \mapsto \Pi_{k=1}^{15} ...

**32**

votes

**7**answers

10k views

### What is the exterior derivative intuitively?

Hi,
actually I have several related questions, not worth opening different threads:
What is the of the exterior derivative intuitively? What is its geometric meaning?
A possible answer I know is, ...

**2**

votes

**2**answers

202 views

### Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...

**5**

votes

**4**answers

786 views

### Levy-Gromov Isoperimetric Inequality

In his paper "Paul Levy's Isoperimetric Inequality", Gromov gives the following isoperimetric inequality:
Let $V$ be a closed $(n+1)$-dimensional Riemannian Manifold with $\mathrm{Ric}(V) \geq n ...

**0**

votes

**1**answer

159 views

### Volume form on pair (X,D)

Let $X$ be a singular Kahler variety with Kahler current $\omega $ then the volume form is $\omega^n$. Now let $D$ be a divisor then how can we define volume form on pair $(X,D) $?

**6**

votes

**1**answer

205 views

### Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...