Some Approaches to Symmetric Spaces

In this study we introduce some approaches , geometrical and algebraic which help to give further understanding of symmetric spaces and help scientists who are seeking for suitable spaces for their applications to carry on their job . One of the aims of this study is to put forward the close connection between different approaches to symmetric spaces , namely algebraic and geometrical features of these spaces with some results . Also giving some conclusions and remarks on the foundation of symmetric spaces with logical ordering of notions and consequences . We think that symmetric space is a very important field for understanding abstract and applied features of spaces , besides they are still need much effort to understand them because of the diversity of their approaches which mix between algebra and geometry , so this Paper is an attempt to disclose some of these features and helps in more understanding and put forward a base for future applications .


Introduction
In studying spaces , one of the aims of this study is to introduce spaces that can suit some scientific applications . Many scientific problems in various fields may have their own conditions that might not agree with the geo metric structure and properties of some spaces familiar to mathematicians and geometers .
Historically the Euclidean space , then the Cartesian and so on have been serving the scientific needs and the solutions to many problems , but mathemat icians always think of new spaces whenever it is crucial to do that , or also they introduce new spaces to be ready for future applications .
In differential geometry , manifolds do their role to meet some scientific demands . This is due to some properties of man ifolds , for instance manifolds need not be connected , or closed . They are never countable unless their dimension is zero .
Differentiable man ifo lds for examp le lead to Lie groups, and these with their Lie algebras help to introduce some abstract spaces such as symmetric spaces .
When we introduce symmetric spaces , this can be done through different approaches of algebraic and geometric nature .
In th is study we are go ing to d iscuss so me o f th ese appro aches wh ich lead to so me invariant p ropert ies of symmet ric spaces . W e can loo k to th ese spaces fro m different aspects which unify some algebraic and geometric A smooth manifold M is a man ifo ld endowed with a smooth structure , that is an atlas of charts satisfying smoothness conditions .
Simp le examp les of smooth manifolds are the Euclidean n-space n  and unit sphere n S .

Proposition
A manifold is locally connected , locally co mpact , a union of a countable collection of compact sets , normal and meterizable .

A Smooth Tensor Field
Let M be a smooth manifo ld . A s mooth tensor field A on the manifo ld M of type ( )

5. Definition ( Riemannian Metric )
A Riemannian metric g on a manifo ld M is a tensor field g :

X Y p is an inner product on
the Tangent space p T M .

Connections and Riemannian Connecti ons
In a s mooth n -man ifo ld M with a smooth Riemann ian metric g , let V be the set of all smooth vector fields on M . A connection on M is the operator , satisfying the conditions : If in addition to the above conditions we have then the connection ∇ is called the Riemannian connection .
We can use for a chart ( ) , U φ on n-man ifo ld : , then using ( ) vi above we find the covariant derivative ( or the connection ) : ,

Parallel Vector Fiel ds & Geodesics
If M is a smooth n -manifo ld with a smooth Riemannian metric g .
Let ∇ be the unique Riemannian connection on M corresponding to g . Let ( ) C t be a smooth regular curve

Riemanni an Manifol d
Let M be a smooth manifold with a metric g , the pair ( M , g ) is called a Riemannian man ifo ld . Geo metric properties of ( M , g ) wh ich only depend on the metric g are called intrin isic or metric properties .

Levi -Ci vita Connecti on
In a Riemann ian manifold with a met ric g , a connection ∇ is called a Levi -Civ ita connection if it is torsion-free , and is compatib le with g , Where :

Lie Groups and Lie Algebras
Lie groups and their Lie algebras are very useful and important tools when studying symmetric spaces , this results from the fact that their algebraic properties derive fro m the group axio ms , and their geometric properties derive fro m the identificat ion of group operation s with points in a topological spaces , and these are man ifo lds .
We should note that every Lie g roup is a smooth man ifold .

Defini tion( Lie Gr oup )
A Lie g roup G is a group satisfying two additional a xio ms : Exa mples: are examples of Lie groups .

Defini tion ( Lie Algebra of a Lie Gr oup )
The tangent space to a linear Lie group G at the identity e , denoted g= e T G is its Lie algebra endowed with a ( non-associative ) mu ltip licat ion , the Lie bracket satisfying the axio ms of a Lie algebra as a vector space .

Homomorphisms of Lie Groups and Lie Algebras
The homo morphis m of the Lie groups 1 2 G and G is a which is a homo mo rphism of groups and a smooth map between the manifo lds 1 2 G and G .
If g 1 and g 2 are two Lie algebras , also the homomo rphism of Lie algebras is a function f : g 1 → g 2 which is a linear map between the vector spaces g 1 and g 2 preserving Lie

Left and Right Translations
If G is a Lie group , for every a G ∈ we define :

Adjoint Representati on
If we denote the Lie group of all bijective linear maps on g by GL(g) , then we call the map a a Ad → where : The derivative : e dAd g → gL(g) denoted by : ad g →gL(g) is called the adjoint representation of g ( where gL(g) denotes the Lie algebra , End(g , g ) of all linear maps on g ) .
In the case of linear group we have According to what is mentioned we can define the Lie algebra of the Lie group as :

Defini tion ( Lie Bracket )
Given a Lie g roup G , g = e T G with the Lie bracket is the Lie algebra of the Lie group G .

Left and Right Invari ant Vector Fiel ds
Given G is a Lie group , we define a vector field ξ on G as : In fact for X ∈ g , we define the vector field is an isomorphis m between G × g and the tangent bundle TG .

Defini tion ( Lie Subalgebra and the Ide al )
If g is a Lie algebra , a subalgebra η of g is a subspace of g ∀ ∈ η and all v ∈ g , we call η an ideal in g . Note that many properties of Lie groups structure can be studied and derived through their Lie algebras , that is why they are important to be studied .
A simple Lie algebra has no proper ideal . The semisimple algebras are constructed of simple ones .

The Exponential Map
In a Riemannian manifold ( , The Lie algebra g generates a group through the exponential mapping . A general group element is exp , , The structure constants define the matrices M of the adjoint representation through ( )

Cartan Subalgebra
In a simp le Lie algebra , we have two kinds of generators : (i) The Cartan subalgebra , which is a maximal abelian subagebra α is an r -dimensional vector called a root , ( ) 1 ,......, r α α α = and r is the rank of the algebra .
The roots form a lattice in the space dual to the Cartan subalgebra . They are useful in the problem o f classification of symmetric spaces .

Killing Form
If g is a Lie algebra , we define the Killing form B of g over afield F as the bilinear form : : The Lie group and its Lie algebra are called semisimp le if the Killing form is nondegenerate .

Homogeneous Space
To introduce homogeneous spaces we should remember the action of the group on a set .

The G-s pace
A locally co mpact Hausdorff space equipped with a transitive action of the group G , is called a G -space .

Theorem
Let G be a locally co mpact group and let X be a transitive G -space . Let 0 x X ∈ and H be the isotropy group of 0 x . If G is co mpact , then given by 0 gH g x → ⋅ is a homo morphis m .

Defini tion ( Homogeneous S pace )
A Homogeneous space X is a transitive G-space that is isomorphic to a quotient space G H . That is , there is an isomorphis m by the mapping Φ as in the previous theorem , ma king G H isomorph ic to X .
In a ho mogeneous space every point looks exactly like every other point . We can also look at a homogeneous space as a space whose isometry group acts transitively on it .

Example
As an examp le of a homogeneous space , the n-sphere

Symmetric Spaces
These are spaces which possess the properties of symmetry and homogeneousness,and they have many applications ,this is because they have mixed algebraic and geometric properties . The beginning for these spaces is that they are spaces with parallel curvature tensor , later they were introduced through different approaches . They have much in co mmon . Any symmetric space has its own special geometry , Euclidean , elliptic and hyperbolic are so me of these geometries .
We give some approaches to symmetric spaces using some algebraic and geometric properties.

Some Approaches to Symmetric S paces
A symmetric space can be considered as : 1 -A Riemannian manifold with point reflect ion . 2 -A Riemannian manifold with parallel curvature tensor . 3 -A Lie group with certain involution . 4 -A homogeneous space with special isotropy group . 5 -A Riemannian manifold with special ho lonomy . 6 -A special Killing vector field . 7 -A Lie triple system . These may be some of many other approaches to symmetric spaces , but we are interested in this work in some of these approaches to reach the required aims of this study .

Symmetric Spaces and Parallel Curvature Tensor
The most famous approach to symmetric spaces is that related to Ellie Cartan , we introduce it here to co mplete the picture of our opinions :

Defini tion
A Riemannian manifold M is called locally symmetric if its curvature tensor R is parallel , that is 0 R ∇ =.

Theorem ( E. Cartan )
Let M be a Riemannian man ifold . Then M is locally symmetric ⇔ 0 R ∇ =, where R is the curvature tensor of M , and ∇ is the connection induced on 4-tensor on M by the Levi -Civita connection of M .

Theorem ( Symmetric Space )
A Co mp lete , Locally symmetric , simp ly connected Riemannian manifold is a symmetric space .
According to what is mentiond above we can give many examples of symmet ric spaces , where we shall give so me details on some of them when introducing other approaches to symmetric spaces . Briefly we can say that

Symmetric Spaces with Point Reflection
Here we consider symmetric spaces as Riemannian man ifolds with point reflections :

Defini tion ( Locally Symmetric Space )
A locally symmetric space is a Riemannian manifold in which the geodesic symmetry at each point is an isommetry in a normal neighbourhood of the point . Sy mmetric Spaces are locally symmetric too, the geodesic symmetries in this case are global iso metries .

Example ( the Unit Sphere )
Let n S be the unit sphere in

Symmetric Space as a Lie Group with Involutive Isometry
This approach can be seen through point reflection , then we can introduce the notion of involution and symmetric pairs with involutive automorphis m .

Defini tion ( Symmetric S pace and Invol uti on )
Let M be a Riemannian man ifold with the metric g such that for every point P M ∈ there exists an isometry P σ of ( ) , M g called an involution such that : Co mposition of involutions will get translations along geodesics which can be used to extend geodesics to the whole of  and this means M is geodesically co mplete .

The Role of Lie Groups
Again we consider Lie groups in this approach , this is through the fact that the group of isometries mentioned above is a Lie group .
By Hopf -Rinow theorem any two points in a geodesically co mp lete Riemannian man ifold can be connected by a geodesic .
We deduce that the translations along the geodesics make the isometry group G ( Lie group ) acting on M transitively . Here also we identify M with G K ( Ho mogeneous space ) , where K is the isotropy group at a point P M ∈ .

Characterization of Symmetric Spaces
One of the most surprising features of symmetric spaces is that we can get their properties and information about them fro m their groups of iso metry and this is another scope for more study .

The Tri ple ( g , η , ρ )
In the symmetric space M , let G be its group of isometry and H is the isotropy group at P M ∈ then G has a Lie group structure and H is a closed Lie subgroup of G . The Lie algebra g of G is the space of Killing vector fields on M . The Lie algebra η of H is a subalgebra of g and has a natural comp lementary subspace ρ such that : g = η ⊕ ρ , [η, η] ⊂ η , [η, ρ ] ⊂ ρ and [ρ , ρ ] ⊂ ρ .
So the characterization of symmet ric spaces is the same as characterizat ion of such triples ( g , η , ρ ) .

Results and Conclusions
→ The geometric and algebraic approaches to symmet ric spaces can be modified to deduce each other . → Most of features of symmetric spaces can be extracted fro m their Lie algebras or the triple ( g , η , ρ ) .
→ The study of symmet ric spaces and continuous research in their properties and classification can lead to most surprising results that can help in their applications .
-The different approaches to symmetric spaces mentioned in this paper , lead to specifying t wo important features of symmetric spaces , that is homogeneousness and symmetry .
Ho mogeneousness can be considered as algebraic property through the transitive action of the isometry group while symmet ry is a geo metric property which can be seen through point reflection in the Riemannian man ifo ld .
An important remark on symmetric spaces is that we can use them in describ ing some abstract and non-abstract spaces , and this unifies the theory and application of spaces which pave the way for future applicat ions in many fields .