An n-dimensional manifold is a second countable Hausdorff topological space such that
- is locally Euclidean: there is and a diffeomorphism is a coordinate system.
- has a differentiable structure: there is a collection of coordinate systems such that:
b. is .
c. is maximal with respect to (b): if is a c.s. such that and are smooth, then .
- Lie Group = Smooth manifold + Topological Group
- Let be a Lie group. If is an abstract closed subgroup of , with the relative topology is a Lie subgroup of ( is a Lie group, ), is a sub-manifold of : is non-singular and is .
- Let be a Lie group. If is closed subgroup of let . Let the natural projection: . Then, has a unique manifold structure such
a. is .
b. There exists local sections of in : if , then there exists and , such that .
above satisfying (a) and (b) is called a homogeneous manifold.
Action of a Lie group on a smooth manifold
An action of on is a – map such that
- Effective, if , there exists such that .
- Transitive, if
Transitive actions of Lie groups on manifolds
Let be a transitive action and let .
- is a closed subgroup of , called the isotropy subgroup at .
- The mapping is a diffeomorphism.
A continuous surjection is a fibration, if has the Homotopy lifting property:
– lifting ,
– There exists to : .
There is a continuous surjection such that has the Homopoty lifting property:
A – vector bundle is a triple where is a surjection such that:
(i) is a -vector space.
(ii) and a homeomorphism is a linear mapping; (ii) says that is trivial in .
(iii) There exists an open covering of there exists , such that with being trivial in .
– Trivial vector bundles:
– Tangent bundle:
Line bundle on the real projective plane
– is called the line bundle of
– is the infinite Möbius band.
Some papers about homogeneous manifolds:
- Kumar V., Ruzhansky M., Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds, Indag. Math., 31 (2020), 266-276. link, arxiv
- Akylzhanov R., Nursultanov E., Ruzhansky M., Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds, J. Math. Anal. Appl., 479 (2019), 1519-1548. arxiv, link
- Daher R., Delgado J., Ruzhansky M., Titchmarsh theorems for Fourier transforms of Holder-Lipschitz functions on compact homogeneous manifolds, Monatsh. Math., 189 (2019), 23-49. arxiv, link (open access)
- Nursultanov E., Ruzhansky M., Tikhonov S., Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 981-1017. arxiv, link
- Dasgupta A., Ruzhansky M., Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bull. Sci. Math., 138 (2014), 756-782. offprint (open access), arxiv, link