Globally symmetric Riemannian space

A Riemannian manifold $ M $ each point $ p $ of which is an isolated fixed point of some involutory isometry $ S _ {p} $ of $ M $, i.e. $ S _ {p} ^ {2} $ is the identity transformation. Let $ G $ be the component of the identity in the group of isometries of the space $ M $ and let $ K $ be the isotropy subgroup at the point $ p $. Then $ M $ is the homogeneous space $ G/K $, and the mapping $ \Phi : g \rightarrow S _ {p} g S _ {p} $ is an involutory automorphism of $ G $; moreover, $ K $ is contained in the closed subgroup $ G ^ \Phi $ of all fixed points of $ \Phi $ and contains the component of the identity in $ G ^ \Phi $.

Let $ g $ be a real Lie algebra, let $ \phi $ be an involutory automorphism of it and let $ k $ be the subalgebra in $ g $ of all $ \phi $- fixed elements. Consider the connected subgroup $ K $ of the associated group $ \mathop{\rm Int} ( g) $ corresponding to the subalgebra $ k $. If the group $ K $ is compact, then $ k $ is called a compactly-imbedded subalgebra of $ g $, while the pair $ ( g, \phi ) $ is known as an orthogonal symmetric Lie algebra. Let $ g = k + m $ be the decomposition into the eigen subspaces of $ \phi $ corresponding to the eigen values 1 and $ - 1 $. The pair $ ( g, \phi ) $ is known as: a) an algebra of compact type if $ g $ is compact and semi-simple; b) an algebra of non-compact type if $ g = k + m $ is a Cartan decomposition; and c) an algebra of Euclidean type if $ m $ is an Abelian ideal in $ g $. Let $ ( g, \phi ) $ be an orthogonal symmetric Lie algebra and let $ g = k + m $ be the above decomposition. Denote by $ g ^ {*} $ the subset $ k + im $ of the complex hull $ g ^ {\mathbf C } $ of $ g $. The mapping

$$ \phi ^ {*} : T + iX \rightarrow T - iX,\ \ T \in k,\ X \in m, $$

is an involutory automorphism of the algebra $ g ^ {*} $, and $ ( g ^ {*} , \phi ^ {*} ) $ is an orthogonal symmetric Lie algebra, called the dual to $ ( g, \phi ) $. If $ ( g, \phi ) $ is an algebra of compact type, then $ ( g ^ {*} , \phi ^ {*} ) $ is an algebra of non-compact type and vice versa.

Each globally symmetric Riemannian space $ G/K $ generates an orthogonal symmetric Lie algebra $ ( g, \phi ) $, where $ g $ is the Lie algebra of the group $ G $ and $ \phi = ( d \Phi ) _ {e} $( $ e $ is the identity in the group). $ G/K $ is called a space of compact or non-compact type, depending on the type of the pair $ ( g, \phi ) $ it generates. All simply-connected globally symmetric Riemannian spaces $ M $ are direct products: $ M = M _ {0} \times M _ {-} \times M _ {+} $, where $ M _ {0} $ is a Euclidean space, and $ M _ {-} $ and $ M _ {+} $ are globally symmetric Riemannian spaces of compact and non-compact types, respectively. For any space of non-compact type the curvature is non-positive in any two-dimensional direction; this curvature is everywhere non-negative for spaces of compact type. Any space of non-compact type is diffeomorphic to a Euclidean space.

Let $ M = G/K $ be a globally symmetric Riemannian space of compact or non-compact type. The rank $ l $ of $ M $ is the maximal dimension of a flat totally-geodesic submanifold in $ M $. Let $ A $ and $ A ^ \prime $ be two flat totally-geodesic submanifolds of $ M $ of dimension $ l $, let $ q \in A $, $ q ^ \prime \in A ^ \prime $, and let $ X $ be the tangent vector to $ M $ at the point $ q $. In such a case: 1) there exists an element $ x \in G $ such that $ xA = A ^ \prime $ and $ xq = q ^ \prime $; and 2) there exists an element $ y \in G $ such that $ yq = q $ and $ d y ( X) $ is the tangent vector to $ A $ at $ q $.

Let $ ( g, \phi ) $ be an orthogonal symmetric Lie algebra and let $ k $ and $ m $ be the eigen subspaces of $ \phi $ corresponding to the eigen values 1 and $ - 1 $. The algebra $ ( g, \phi ) $ is called irreducible if the following conditions are satisfied: 1) $ g $ is a semi-simple algebra and $ k $ does not contain non-zero ideals of $ g $; and 2) the algebra $ \mathop{\rm ad} _ {g} ( k) $ acts irreducibly on $ m $. A globally symmetric Riemannian space $ G/K $ is called irreducible if the orthogonal symmetric Lie algebra $ ( g, \phi ) $ generated by $ G / K $ is irreducible. Two orthogonal symmetric Lie algebras $ ( g, \phi ) $ and $ ( g ^ \prime , \phi ^ \prime ) $ are called isomorphic if there exists an isomorphism $ \psi $ of the algebra $ g $ onto $ g ^ \prime $ such that $ \psi \circ \phi = \phi ^ \prime \circ \psi $. The classification of simply-connected irreducible globally symmetric Riemannian spaces up to an isometry is equivalent to the classification of irreducible orthogonal symmetric Lie algebras up to an isomorphism.

The irreducible orthogonal symmetric Lie algebras of compact type are: I. $ ( g, \phi ) $, where $ g $ is a compact simple Lie algebra and $ \phi $ is any of its involutory automorphisms; II. $ ( g, \phi ) $, where the compact algebra $ g $ is a direct sum of two simple ideals that are conjugate through the automorphism $ \phi $.

The irreducible orthogonal symmetric Lie algebras of non-compact type are: III. $ ( g, \phi ) $, where $ g $ is a simple non-compact Lie algebra over $ \mathbf R $ whose complex hull $ g ^ {\mathbf C } $ is a simple Lie algebra over $ \mathbf C $ and $ \phi $ is an involutory automorphism of $ g $ such that its fixed points constitute a maximal compactly-imbedded subalgebra; IV. $ ( g, \phi ) $, where $ g $ is a simple Lie algebra over $ \mathbf C $ considered as a real Lie algebra and $ \phi $ is the conjugation of $ g $ with respect to a maximal compactly-imbedded subalgebra $ k $, i.e. a mapping $ X + iY \rightarrow X - iY $, $ X, Y \in k $. Moreover, if $ ( g ^ {*} , \phi ^ {*} ) $ denotes the algebra dual to $ ( g, \phi ) $, then the latter is of type III or IV if $ ( g ^ {*} , \phi ^ {*} ) $ is, respectively, of type I or II, and vice versa.

Only one globally symmetric Riemannian space is connected with each irreducible orthogonal symmetric algebra of non-compact type, and this space is simply connected. As regards compact algebras, the solution of the corresponding problem is much more complicated. It is sufficient to consider globally symmetric Riemannian spaces of type I and type II, connected with algebras of type II — these are in fact connected compact simple Lie groups endowed with a Riemannian structure which is invariant under left and right translations. The problem of classifying globally symmetric Riemannian spaces connected with algebras of type I, up to local isometries, is equivalent to the problem of classifying involutory automorphisms of simple compact Lie algebras. The global classification of the symmetric Riemannian spaces connected with a given orthogonal symmetric algebra $ ( g, \phi ) $ of compact type is solved by the following theorem.

Let $ ( g, \phi ) $ be an orthogonal symmetric algebra of compact type, where the subalgebra $ k $ of fixed points of $ \phi $ contains no ideals of $ g $ other than $ \{ 0 \} $. Let $ \widetilde{G} $ be a simply-connected Lie group with Lie algebra $ g $, let $ \widetilde{Z} $ be the centre of $ \widetilde{G} $, let $ \widetilde \Phi $ be an automorphism of $ \widetilde{G} $ such that $ d \widetilde \Phi = \phi $, and let $ \widetilde{K} $ be the set of fixed points of $ \Phi $. For an arbitrary subgroup $ S $ of $ \widetilde{Z} $ one puts $ K _ {s} = \{ {g \in \widetilde{G} } : {g ^ {-} 1 \Phi ( g) \in S } \} $. The globally symmetric Riemannian spaces $ M $ connected with $ ( g, \phi ) $ coincide with the spaces of the form $ G/K $ where $ G = \widetilde{G} / S $, $ K = K ^ {*} /( K ^ {*} \cap S) $, endowed with an arbitrary $ G $- invariant metric. Here $ S $ runs through all subgroups of $ \widetilde{Z} $, while $ K ^ {*} $ runs through all subgroups of $ \widetilde{G} $ for which $ K \subset K ^ {*} \subset K _ {s} $.

References

[1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2]	O. Loos, "Symmetric spaces" , 1–2 , Benjamin (1969)