Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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3
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0answers
34 views

Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
1
vote
0answers
66 views

Sobolev space properties and trace on a non-compact Riemannian manifold with boundary

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$. I am ...
2
votes
1answer
199 views

Distance function to a submanifold

Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where ...
1
vote
1answer
161 views

Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles

Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl_p(\mathbb CP^2)$, then for the Line bundle $L=-K_X$, we have for Futaki invariant $Fut_L\neq 0$, I want to see, what about ...
1
vote
2answers
148 views

Dual connections for Information Geometry

In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied ...
0
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0answers
38 views

Is the blowing up the rectifiable set cone?

Let $M$ be a rectifiable set in $\mathbb{R}^N$. For any point $p\in M$, is the following true?: $\lambda_i M$ subconverges to a cone in $p$ for $\lambda_i\to\infty$, i.e. $(\lambda_i M)\cap B(p,R)$ ...
0
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0answers
90 views

A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article. Formulation: Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...
4
votes
1answer
293 views

Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set of such a polynomial gives a real curve ...
2
votes
1answer
98 views

A surface on which all regular curves have nowhere vanishing curvature

Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that ...
4
votes
1answer
128 views

Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$. Q. Does every non-closed geodesic $\gamma$ fill $P$ densely? Of course $\gamma$ cannot pass through a vertex of $P$, but ...
1
vote
0answers
95 views

Continuous family of constant scalar curvature metrics

The question is as follows: Does there exist an example of a (continuous) family of metrics $g_t$ on a compact manifold such that the following properties hold? All metrics $g_t$ have constant ...
0
votes
0answers
36 views

second fundamental form of boundary of convex subset non-negative? [closed]

Let M be an n-dim non-compact Riemannian manifold without boudary and $Ric\geqslant-(n-1)$. Let $Y$ be a convex subset of M with smooth boundary, then the second fundamental form of the boundary ...
2
votes
0answers
112 views

Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
6
votes
2answers
262 views

Moduli spaces of connections as representation spaces

It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$. Furthermore, Atiyah and Bott (1983) showed ...
0
votes
1answer
146 views

floer homology and viterbo's theorem

Let $M$ be a compact manifold. In their paper "On the Floer Homology of Cotangent Bundles", A. ABBONDANDOLO and M. SCHWARZ define the Floer homology of $T^*M$ by looking at 1-periodic Hamiltonian ...
5
votes
3answers
421 views

Using Stokes' theorem to define “area” enclosed by a curve

I am trying to figure out what the next calculation of the "area" (or "volume" in higher dimensional analogues) using Stokes' theorem really means. Here is my thought process: $2$-dimensional case: ...
2
votes
1answer
91 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
0
votes
1answer
101 views

Surjectivity of “nice maps” from local properties

What tools are available from real algebraic geometry, analysis and topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$ from local properties and maybe function values? ...
0
votes
2answers
130 views

Frobenius condition

Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...
2
votes
2answers
80 views

Estimates on a heat process with fixed boundary data and zero initial conditions

Consider the following heat process: For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that $$ \partial_t ...
1
vote
2answers
287 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 ...
0
votes
1answer
89 views

Complex transport equation

Consider an n dimensional Riemannian manifold with boundary. Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...
1
vote
0answers
92 views

How to find a invariant surface of a diffeomorphism

Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows: $$T(x,y,z)=(y,z,yz-x).$$ We can calculated that $T$ has the folliwng ...
0
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0answers
76 views

Evolution of local oscillation of scalar curvature under Ricci flow

I apologize in advance if the question will turn out to have an obvious answer but my knowledge of Ricci flow is quite limited. Let $(M,g)$ be a smooth compact Riemannian manifold. I denote by $d_{g}$ ...
12
votes
4answers
532 views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
3
votes
1answer
101 views

Surfaces with specific types of second fundamental form

Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...
5
votes
2answers
135 views

Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
2
votes
1answer
153 views

Eigenfunction on surface with boundary

Suppose we have a two-dimensional surface $M$ with smooth boundary $\partial M$. Equip $M$ with a metric $g$ such that the Gauss curvature $K$ of $M$ and geodesic curvature $\kappa$ of $\partial M$ ...
0
votes
0answers
82 views

Closed Invariant Forms on Complex Projective $k$-Space

Considering complex projective $k$-space as the homogeneous space $SU_k/U_{k-1}$, is it true that every $SU_k$-invariant form is closed?
4
votes
0answers
192 views

Are there Zoll pancakes?

How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be? I don't want to impose a ...
2
votes
2answers
267 views

Geometrical interpretation of a Schrödinger operator

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
1
vote
0answers
52 views

Besicovitch's covering theorem for ellipsoids and shadows

The usual Besicovitch's covering theorem concerns closed balls in $\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there ...
9
votes
1answer
686 views

Formula for the Perimeter of a spherical triangle?

Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given ...
0
votes
0answers
84 views

kahler manifolds with positive holomorphic sectional curvature

It is well known that a compact Kahler manifold with positive holomorphic bisectional curvature is biholomorphic to $CP^n$. However, if we just assume positive holomorphic sectional curvature, is ...
5
votes
2answers
219 views

Poincare-like inequality on compact Riemannian manifolds

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of $$ ...
2
votes
1answer
376 views

Symplectic quotient of projective variety is projective?

Let $G$ be a compact connected Lie group and $\mathfrak g^*$ be dual of Lie algebra $\mathfrak g$. Let $M$ be a compact projective variety and $G$ act on $M$ freely and $M$ is $G$ equivariant, and ...
1
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0answers
121 views

Electrodynamics modelled by U(1) gauge theory [closed]

As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
4
votes
1answer
108 views

Minimal surfaces + Semi-Geodesic Coordinates

Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
3
votes
2answers
192 views

Isothermal-related functions in higher dimensions

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help. Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
1
vote
2answers
145 views

Handle body of 3-manifold with boundary

We know from Morse theory that smooth manifold(with or without boundary) is a handlebody. However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...
0
votes
2answers
172 views

Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
0
votes
1answer
196 views

Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...
0
votes
1answer
89 views

A decomposition of incompressible vector fields

In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved: Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$. Call $w$ its ...
5
votes
1answer
125 views

Averaging maps of Riemannian manifolds

Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ...
0
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0answers
79 views

Derivability of a function defined on the tangent bundle. Foundations of Finsler metrics

My question is linked to the foundations of Finsler metrics (with weak derivability assumptions). Let $M$ be a manifold of dimension $n$, and $F$ is a function from the tangent bundle $TM$ to ...
2
votes
0answers
89 views

“simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...
8
votes
2answers
418 views

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...
0
votes
1answer
99 views

Normals along a Sphere [closed]

Let $M \subset \mathbb{R}^d$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) ...
19
votes
0answers
251 views

Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$. This is a not too difficult theorem due to Whitney, proved in many textbooks. ...
3
votes
1answer
55 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...