### Smooth Manifold

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:

a. .

b. is .

c. is maximal with respect to (b): if is a c.s. such that and are smooth, then .

### Lie groups

- Lie Group = Smooth manifold + Topological Group

### Homogeneous manifolds

- 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.
- .

**Examples:**

**Hopf fibration:**

**Fibration **

A continuous surjection is a fibration, if has the Homotopy lifting property:

–

– lifting ,

– There exists to : .

**Hopf Fibration**.

There is a continuous surjection such that has the Homopoty lifting property:

## Vector Bundles.

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 .

**Examples:**

– Trivial vector bundles:

– Tangent bundle:

**Examples:**

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,*, 31 (**Indag. Math.***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*,, 479 (*J. Math. Anal. Appl.**2019*), 1519-1548. arxiv, link - Daher R., Delgado J., Ruzhansky M.,
*Titchmarsh theorems for Fourier transforms of Holder-Lipschitz functions on compact homogeneous manifolds*,, 189 (*Monatsh. Math.**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,*, 16**Ann. Sc. Norm. Super. Pisa Cl. Sci.***(2016),*981-1017. arxiv, link - Dasgupta A., Ruzhansky M.,
*Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces,*138**Bull. Sci. Math.**,*(2014),*756-782*.*offprint (open access), arxiv, link