# Tagged Questions

37 views

### geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
68 views

### Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-))$$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
53 views

### All differentiable functions on $\mathbb{S}^n$

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 ...
71 views

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
65 views

103 views

### Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
62 views

### French translation, and what is the curvature of a metric?

I have a french paper to read. There is the notion of une collection des courbures des métriques $g_t$. Now I would guess that this refers to a collection of curvatures of metrics $g_t$, however ...
214 views

### Covert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels x units in Euclidean space, how much ...
57 views

### Boundary Normal Coordinates

Let $(M,g)$ be a 2D Riemannian manifold with boundary. The boundary normal coordinates $\psi$ are constructed roughly as follows: in a sufficiently small neighborhood $U$ of $\partial M$, for ...
68 views

60 views

### A positive constant don't change the Levi-Civita Connection

In the Chow's book there is a question that I can't solve, the question is Let $\nabla^g$ denote the Levi-Civita connection of the metric $g$. Show that for any constant $c>0$ and metric $g$, ...
78 views

### When does a Green's function exist?

If I have a simply-connected compact domain $\Omega$ in $\mathbb{R}^2$, endowed with a Riemannian metric $g$, does there exist a green's function on $\Omega$ for the laplace operator induced ...
53 views

### pre-quantization on cotangent bundle $T^*M$

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?
79 views

### If $M$ is complete is the closed ball compact?

Let $M$ be a Riemannian manifold and $p,q \in M$. Let $\Omega=\Omega(M;p,q)$ be the set of piecewise $C^\infty$ paths from $p$ to $q$. Let $\rho$ denote the topological metric on $M$ coming from its ...
49 views

### Tangent space of loop space.

Let $\Omega$ be the path space of a riemannian manifold $M$. I have to define the tangent space of $\Omega$ in a path $\omega$, that I denote with $T_p \Omega$. I think that this space is the vector ...
46 views

### isomorphism of two compex line bundles

I am looking for some non-trivial examples of Line Bundles and an example about isomorphism of two line bundles. With details
90 views

### Second variation formula and Jacobi fields

Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation od a geodesic $\gamma$. For $i=1,2$ we define $$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$ the ...
50 views

149 views

### Volume element of the Sphere

If we consider the sphere on $E^3$ with Riemannian metric $G=dx + dy + dz$ then transforming to spherical coordinates we get $G=R^2 d{\theta} +R^2 sin(\theta) d{\phi}^2$. Hence the volume form is ...
73 views

### Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
201 views

### Curvature of a metric defined on an open disc in $\mathbb{R}^2$

Let $D$ be an open disc centred at the origin in $\mathbb{R}^2$. Give $D$ a Riemannian metric of the the form $(dx^2+dy^2)/f(r)^2$, where $r=\sqrt{x^2+y^2}$ and $f(r)>0$. Show that the curvature of ...
111 views

### Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
250 views

### Is there a non-variational derivation of Snell's law from Fermat's principle?

Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
63 views

### Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
113 views

### Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
85 views

### Geodesics (1): Spaces with more than two geodesics between two points

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
71 views

### Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
154 views

### Given lattice G; parameters of torus R^2/G?

This should be a simple, known result, but I can't seem to find it. Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and ...
55 views

### Relation with Jacobi fields in a small neighbourhood of some point in a complete manifold

I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
296 views

### Most succinct proof of the uniqueness and existence of the Levi-Civita connection.

Seeing as proving the existence and/or uniqueness of the Levi-Civita connection seems to crop up in every single exam in Geometry and General Relativity, what is the most succinct proof of this, to ...
320 views

### Notation for covariant derivative

I'm reading John M. Lee's book " Riemannian Manifolds". On page 57, the covariant derivative of $V$ along a curve $\gamma$ is defined, where $V$ is a vector field along $\gamma$. It is denoted by ...
158 views

### A semi-Riemannian geometry exercise

How can I prove that there are no compact semi-Riemannian hypersurfaces in semi-euclidean space $\mathbb{R}_v^n$ of index $v$ with $0<v<n$??. Thanks for any help!!
If I start with an infinite flat sheet of graph paper, and in polar coordinates cut out a piece according to: $r>0, \ \ -f(r) < \theta < f(r)$ Now I want to stitch the remaining graph paper ...