1
vote
0answers
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. ...
4
votes
1answer
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 ...
1
vote
0answers
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 ...
1
vote
1answer
71 views

Reference about Gauss-Bonnet-Chern theorem.

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 ...
0
votes
1answer
65 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
1
vote
0answers
27 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
1
vote
0answers
41 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
2
votes
1answer
82 views

Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
1
vote
1answer
62 views

Find the Gauss Curvature of This Particular Metric:

Let D be an open disc centred at the origin in $ \Bbb R^2 $. Give D a Riemannian metric of the form $ (dx^2 + dy^2)/f(r)^2 $, where $ r = \sqrt{x^2 + y^2} $ and $ f(r) > 0 $. Show that the Gauss ...
2
votes
1answer
205 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
0
votes
1answer
54 views

How to find a dual frame at the sphere

Let $U=\{ (\theta,\phi,r):\theta \in \mathbb{R}, \phi \in ]0,\pi[,r\gt 0\}$, how I can find the moving frame. I thougt: Consider the parametization for $U$ $$(\theta,\phi,r)\mapsto ...
1
vote
1answer
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
0
votes
0answers
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
3
votes
0answers
68 views

Distance function and Green's function

Let $ (M,g) $ be a two dimensional Riemmanian manifold, with a smooth distance function $ d(x,y) $ for all $ x, y$ in $ M$. The logarithm of the distance function then satisfies $ \Delta \left( ...
0
votes
0answers
82 views

Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the ...
1
vote
1answer
45 views

Derivative of tangent vetor

Let $x(\theta,\phi)=(\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$ be the standard local coordinates for $\mathbb{S}^{2} \setminus \lbrace (0,0,\pm 1) \rbrace$. Let $\phi_{0} \in ...
1
vote
1answer
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 $g$, ...
2
votes
0answers
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 ...
0
votes
1answer
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?
0
votes
1answer
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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
1
vote
0answers
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 ...
0
votes
1answer
50 views

Path spaces and induced maps on tangent spaces

Let $M$ be a smooth manifold and $\Omega(M)$ the set of all piecewise smooth path in $M$. Let be $$ f: \Omega(M) \rightarrow \mathbb{R} $$ How can I define $$ f^*: T_{\omega}\Omega \rightarrow ...
2
votes
0answers
72 views

angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
1answer
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 ...
3
votes
2answers
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 ...
3
votes
1answer
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
3
votes
0answers
71 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
1
vote
1answer
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 ...
0
votes
0answers
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 ...
7
votes
1answer
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 ...
3
votes
2answers
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 ...
1
vote
1answer
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!!
1
vote
0answers
343 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
2
votes
1answer
116 views

Stitching together piece of flat space

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 ...