**3**

votes

**1**answer

224 views

### Taylor expansion of the determinant of a Riemannian metric

Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...

**0**

votes

**1**answer

85 views

### Orientability of Stiefel manifold V2(R4) [closed]

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups ...

**1**

vote

**0**answers

78 views

### A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of Schrödinger operator is simple 1 ...

**1**

vote

**0**answers

103 views

### Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...

**0**

votes

**1**answer

185 views

### Describe all differentiable functions on $\mathbb{S}^n \backslash S$ (S is the south pole) [closed]

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...

**2**

votes

**1**answer

122 views

### local approximation of a vector field on a Riemannian manifold

Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f_i$, such that the ...

**2**

votes

**3**answers

347 views

### If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which
acts on $M$ and preserves complex structure then If $M$ has Kaehler
manifold, then the symplectic quotient of $M$, i.e, ...

**4**

votes

**1**answer

151 views

### Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions)

Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I ...

**2**

votes

**0**answers

90 views

### Chern-Weil theory for degenerated metric

If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of ...

**8**

votes

**1**answer

611 views

### Learning higher differential geometry

I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the ...

**0**

votes

**0**answers

64 views

### decomposition in cohomology from the antipodal map

Consider the antipodal map $i:S^{n}\to S^{n}$, i.e $i:x\to-x$, its
induces a decomposition $\Omega^{n}(S^{n})=\Omega^{n}_{+}(S^{n})\oplus\Omega^{n}_{-}(S^{n})$, where $\Omega^{n}_{\pm}(S^{n})$ ...

**6**

votes

**1**answer

196 views

### cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial
and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...

**7**

votes

**1**answer

436 views

### Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know
Let $(M,\omega)$ be a compact ...

**0**

votes

**0**answers

60 views

### Local behavior of Killing spinor on Sasaki-Einstein Manifold

I am trying to understand how a Killing spinor behaves near a closed Reeb orbit, for instance, on $S^5$ and $Y_{p,q}$ manifolds
So Let us consider the Killing spinor equation on a five-dimensional ...

**2**

votes

**1**answer

93 views

### Implicit function theorem for operator

I am reading the paper of Convergence of the Yamabe flow for arbitrary initial energy
I am stuck by one part of the paper. Suppose $u_\infty>0$ is a smooth function on $(M, g_0)$ and
...

**2**

votes

**1**answer

81 views

### Eigenvalue problem of an operator involving the exterior derivative of differential forms

Consider two functions $\alpha,\beta: \mathbb{R}^2 \to \mathbb{R}$, where $\alpha$ is given and we look for solutions $\beta$ such that
$$*(d\alpha \wedge d\beta) = \lambda \beta$$
for some $\lambda ...

**0**

votes

**0**answers

79 views

### Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian?

Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian?
If possible I'd also like to know if the right translation of the Shatten $p-norm$ on the Lie algebra gives rise to a ...

**4**

votes

**4**answers

323 views

### Example of Non-Conformally Flat Einstein Manifold?

Does there exist an Einstein manifold which is not conformally flat, which is to say one which has non-vanishing Weyl tensor. If so, what is a good example.

**1**

vote

**1**answer

157 views

### When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and
$\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...

**0**

votes

**0**answers

37 views

### harmonic functions on hyperbolic plane x Real line

I am looking for references of harmonic functions(no growth condition) on the product of hyperbolic plane $\mathbb H^2$ with the real line $\mathbb R$. The metric is just the product one. Especially ...

**7**

votes

**1**answer

417 views

### A complete Riemannian metric for which the Ricci flow has no solution

What is an example of a connected complete Riemannian manifold such that the Ricci flow has no solution with the given Riemannian metric as initial data? As Terry Tao points out, it is easy to ...

**1**

vote

**1**answer

79 views

### Pseudohermitian Structures and Contact Metric Structures

Let $M$ be an odd-dimensional manifold. An almost contact metric structure on $M$ is a 4-tuple $(\xi, \eta, \phi, g)$, where $\xi$ is a vector field, $\eta$ a one-form, $\phi$ an endomorphism of the ...

**3**

votes

**1**answer

144 views

### Extension of pseudodifferential operators

I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( ...

**1**

vote

**1**answer

231 views

### Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement:
Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be ...

**1**

vote

**1**answer

113 views

### characterization of structure group

Somebody tell me that:
For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power.
Assume that the ...

**0**

votes

**0**answers

67 views

### Ring structure for $K^{-1}$?

My questions are
whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say.
If such a ring structure ...

**1**

vote

**0**answers

171 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 ...

**1**

vote

**1**answer

123 views

### Natural bundle, g-natural metric, meaning

I am trying to understand meaning and importance of a g-natural metric. Since I do pure differential geometry for my research, I am not familiar with many notions which are needed for understanding a ...

**0**

votes

**0**answers

87 views

### Application of conformal normal coordinates for higher order elliptic operator

Let $n>2$ be even. Consider a compact Riemannian manifold $(M^n,g)$ and denote with $P_g$ the critical GJMS operator.
Recall that $P_g$ is conformally invariant, i.e.
$$P_{\tilde g}=e^{-nu}P_g$$ ...

**1**

vote

**1**answer

175 views

### examples of Kähler manifolds with trivial Hodge numbers and first Chern classes

Yesterday I asked the following question to which abx has given a positive answer.
examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes
But I suddenly realized ...

**0**

votes

**1**answer

293 views

### A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, ...

**2**

votes

**1**answer

247 views

### examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and ...

**1**

vote

**1**answer

152 views

### de Rahm Laplace operator on forms bounded

Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...

**5**

votes

**1**answer

290 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**8**

votes

**1**answer

180 views

### Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...

**3**

votes

**0**answers

303 views

### Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...

**3**

votes

**0**answers

165 views

### Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I ...

**4**

votes

**0**answers

81 views

### Is this groupoid a model for the derived fixed-point locus of the free loop space?

In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...

**4**

votes

**1**answer

200 views

### What is an element of an iterated tangent bundle?

An element of the tangent bundle $T M$ of a manifold is called a "(tangent) vector". An element of its dual $T^* M$ is called a "covector" or a "1-form". An element of the exterior square ...

**6**

votes

**1**answer

167 views

### Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, thus
$$3\tau+2\chi=c_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...

**1**

vote

**2**answers

352 views

### Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$,
i.e., each $p=(p(x_0),\dots,p(x_n))\in ...

**1**

vote

**1**answer

96 views

### Injectivity radius on complete manifolds with curvature decay

I am wondering that the following statement is true or not:
Let $(M,g)$ be a complete non-compact Riemannian manifold with $0 < Sect \leq C\cdot dist(O,x)^{-2a}$, $a\in(0,1]$. ($O$ is a point ...

**11**

votes

**3**answers

744 views

### Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...

**1**

vote

**1**answer

357 views

### Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function).
I am looking for the solutions among of ...

**0**

votes

**1**answer

104 views

### both convex and superharmonic function on manifold concave?

M is a non-compact Rimannian manifold without boundary.
$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e.
$$
-\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M ...

**7**

votes

**2**answers

334 views

### Riemannian distance functions on the real line

A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties:
$d$ is a length metric ...

**0**

votes

**0**answers

39 views

### Expansion of the squared distance function from a submanifold

let $(M,g)$ be a compact Kahler manifold of dimension $m$ with the Kahler metric $g$ real-analytic. Let $N\subset M$ be a complex submanifold of dimension $0<k<m$. Let $$d_{g}(M,\cdot): ...

**3**

votes

**1**answer

137 views

### Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?

Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...

**0**

votes

**0**answers

74 views

### Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form:
Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...

**1**

vote

**2**answers

106 views

### Right invariant Killing fields of Right invariant Riemanian metrics

Can there exist a right invariant killing field of a right invariant (but not bi-invariant) Riemannian metric on a Lie group?
I am especially interested in the case of $SU(N)$ with a metric of the ...