All Questions
13 questions
3
votes
1
answer
100
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
2
votes
0
answers
108
views
Questions about symmetric spaces
I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
- 563
5
votes
0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
9
votes
1
answer
444
views
Compact flat orientable 3 manifolds and mapping tori
There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are ...
- 4,081
3
votes
1
answer
425
views
3 dimensional solvmanifolds and Thurston geometries
Does every three dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?
definitions/motivation/background:
A solvmanifold is a manifold $ M $ admitting a transitive action by a ...
- 4,081
2
votes
1
answer
484
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 4,081
6
votes
0
answers
341
views
When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
- 5,405
4
votes
1
answer
230
views
Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi_1(\Sigma) $ of $ \SL_2(\...
- 4,081
8
votes
1
answer
599
views
Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
- 4,081
2
votes
1
answer
137
views
noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous
Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
- 4,081
2
votes
2
answers
213
views
Riemannian homogeneous equivalent to linear group orbit
Let $ M $ be a smooth manifold.
Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $.
Recall that a manifold $ M $ is Riemannian homogeneous if ...
- 4,081
7
votes
1
answer
546
views
Can a hyperbolic manifold be a product?
I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
- 19.3k
4
votes
1
answer
233
views
Flat solvmanifolds?
I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
- 481