**1**

vote

**2**answers

69 views

### Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$

Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...

**1**

vote

**1**answer

130 views

### Poincare inequality on balls to arbitrary open subset of manifolds

Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$
$$
\frac{1}{m(B)}\int_B ...

**1**

vote

**1**answer

148 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 ...

**0**

votes

**1**answer

126 views

### General description of surface with zero gaussian curvature

Let suppose function $g(s,t)$ satisfies partial differential equations:
$g_{ss} g_{tt} - g_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution ...

**1**

vote

**0**answers

119 views

### Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...

**4**

votes

**2**answers

178 views

### Projective curves of constant curvature

A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...

**1**

vote

**0**answers

70 views

### Determine the shape of curve which has the minimum number of inflection points

Let $A$ be a set of generic $C^2$ closed curves in $2$-dimentional Euclidean space, and an equivalence relation between $2$ curves $a$ and $b$ in $A$ is defind that there exist diffeomorpfic map $h ...

**2**

votes

**1**answer

105 views

### Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation
$$\frac{\partial f ...

**0**

votes

**0**answers

43 views

### Elliptic operator on unit tangent bundle and positive sectional curvature

In "compact kahler manifolds with nonnegative sectional curvature" ( invent math, 1977),
Gray introduced a differential operator $L$ on the unit tangent bundle on a Riemannian manifold $(M, g)$. It ...

**2**

votes

**0**answers

122 views

### Conjugate or focusing points on null geodesics imply chronality

Theorem
Let $\beta\colon [0,1] \to M$ be a null geodesic. If $\beta(t_0)$ is conjugate to $\beta(0)$ along $\beta$ for some $t_0\in (0,1)$, then there is a timelike curve from $\beta(0)$ to ...

**8**

votes

**0**answers

140 views

### Commutative spectral triples not coming from manifolds

There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...

**5**

votes

**0**answers

94 views

### (Un)bounded Geometry and Sobolev Spaces

This post is related to this and this post.
It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < ...

**4**

votes

**0**answers

89 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 ...

**2**

votes

**0**answers

133 views

### Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

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 ...

**4**

votes

**2**answers

320 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

**1**answer

192 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

**2**answers

165 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**

votes

**0**answers

42 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**

votes

**0**answers

100 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

**1**answer

306 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

**1**answer

107 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 ...

**5**

votes

**1**answer

147 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 ...

**2**

votes

**0**answers

105 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 ...

**2**

votes

**0**answers

125 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

**2**answers

276 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

**1**answer

166 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

**3**answers

438 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

**1**answer

103 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

**1**answer

112 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

**2**answers

138 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

**2**answers

86 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

**2**answers

381 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

**1**answer

94 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

**0**answers

97 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**

votes

**0**answers

93 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

**4**answers

588 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

**1**answer

109 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

**2**answers

172 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

**1**answer

157 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

**0**answers

88 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

**0**answers

195 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

**2**answers

279 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

**0**answers

68 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

**1**answer

739 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

**0**answers

102 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

**2**answers

246 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

**1**answer

382 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**

vote

**0**answers

138 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

**1**answer

119 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

**2**answers

200 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 ...