1
vote
0answers
17 views

Circle action on the product of a Mobius band and a circle.

Consider the product of a Möbius band and a circle $Mo\times S^1$. Is there a circle action on $Mo\times S^1$ such that it is equivariantly homeomorphic to the twisted product $D^2 ...
0
votes
0answers
14 views

How can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space..

how can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space? Thanks..
1
vote
0answers
59 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
0
votes
1answer
43 views

Prove this action is properly discontinuous..

Consider the group $\mathbb Z_2=\{0, 1\}$ acting on the sphere $\mathbb S^n$ through the group actions $\psi_0=Id$ e $\psi_1=-Id$. Show this actions is properly discontinuos? The definition of ...
2
votes
2answers
102 views

How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?

Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by ...
1
vote
1answer
42 views

How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$?

How to show the orbit space $\mathbb S^n/\mathbb Z_2$ is $\mathbb RP^n$? Here $\mathbb Z_2=\{0, 1\}$ is the additive group and the group action considered induces the aplications $\psi_0=Id$ and ...
2
votes
1answer
81 views

Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
0
votes
0answers
107 views

Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up: Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...
1
vote
0answers
47 views

A K-invariant submanifold of G-manifold and fundamental vector fields

Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$ $$ \underline{X}(p) := ...