Invariant metric

A Riemannian metric $ m $ on a manifold $ M $ that does not change under any of the transformations of a given Lie group $ G $ of transformations. The group $ G $ itself is called a group of motions (isometries) of the metric $ m $( or of the Riemannian space $ ( M , m ) $).

A Lie group $ G $ of transformations of a manifold $ M $ acting properly on $ M $( that is, the mapping $ G \times M \rightarrow M \times M $, $ ( g , x ) \rightarrow ( gx , x ) $ is proper) has an invariant metric. Conversely, the group of all motions of any Riemannian metric (as well as any closed subgroup of it) is a proper Lie group of transformations. In this case the stabilizer (or isotropy group)

$$ G _ {x} = \{ {g \in G } : {gx = x } \} $$

of any point $ x \in M $ is a compact subgroup of $ G $. If $ G $ itself is compact, then a $ G $- invariant metric $ m _ {0} $ can be constructed on $ M $ by averaging any metric $ m $ on $ M $ over $ G $:

$$ m _ {0} = \int\limits _ { G } ( g ^ {*} m ) dg , $$

where the integral is taken with respect to the Haar measure.

If $ G $ is transitive, $ M $ can be identified with the space of cosets $ G / H $ of $ G $ with respect to the stabilizer $ H = G _ {x _ {0} } $ of a fixed point $ x _ {0} \in M $, and in order that there exist a $ G $- invariant metric on $ M $ it is necessary and sufficient that the linear isotropy group (see Isotropy representation) has compact closure in $ \mathop{\rm GL} ( T _ {x _ {0} } M ) $( in particular, it is sufficient that $ H $ be compact). In this case the space $ G / H $ is reductive, that is, the Lie algebra $ \mathfrak G $ of $ G $ admits a decomposition $ \mathfrak G = \mathfrak H + \mathfrak M $, where $ \mathfrak H $ is the subalgebra corresponding to $ H $ and $ \mathfrak M $ is a subspace that is invariant under $ \mathop{\rm Ad} H $ where $ \mathop{\rm Ad} $ is the adjoint representation of $ G $( cf. Adjoint representation of a Lie group). If $ \mathfrak M $ is identified with $ T _ {x _ {0} } M $, then any $ G $- invariant metric $ m $ on $ M $ is obtained from some $ \mathop{\rm Ad} H $- invariant Euclidean metric $ \langle , \rangle $ on $ \mathfrak M $ in the following way:

$$ m _ {x} ( X , Y ) = \langle ( g ^ {*} ) ^ {-1} X, ( g ^ {*} ) ^ {-1} Y \rangle ,\ \ X , Y \in T _ {x} M , $$

where $ g \in G $ is such that $ g x _ {0} = x $.

The tensor fields associated with a $ G $- invariant metric (the curvature tensor, its covariant derivatives, etc.) are $ G $- invariant fields. In the case of a homogeneous space $ M = G / H $, their value at a point $ x _ {0} $ can be expressed in terms of the Nomizu operator $ L _ {X} \in \mathop{\rm End} ( \mathfrak M ) $, which is defined by the formula

$$ L _ {X} Y = - \nabla _ {Y} X ^ {*} = \ ( {\mathcal L} _ {X ^ {*} } - \nabla _ {X ^ {*} } ) _ {x _ {0} } Y ,\ \ Y \in \mathfrak M ,\ X \in \mathfrak G , $$

where $ X ^ {*} $ is the velocity field of the one-parameter group of transformations $ \mathop{\rm exp} tX $, $ \nabla $ is the covariant differentiation operator of the Riemannian connection and $ {\mathcal L} $ is the Lie derivative operator. In particular, the curvature operator $ \mathop{\rm Riem} ( X , Y ) $ and the sectional curvature $ K ( X, Y ) $ in the direction given by the orthonormal basis $ X , Y \in M $ satisfy the following formulas:

$$ \mathop{\rm Riem} ( X , Y ) = [ L _ {X} , L _ {Y} ] - L _ {[ X , Y ] } , $$

$$ K ( X , Y ) \equiv - \langle \mathop{\rm Riem} ( X , Y ) X , Y \rangle = $$

$$ = \ \langle L _ {X} Y , L _ {X} Y \rangle - < [ X , Y ] _ {\mathfrak M } , [ X , Y ] _ {\mathfrak M } > + $$

$$ - \langle [ Y , [ Y , X ] ] _ {\mathfrak M } , X \rangle - \langle L _ {X} X , L _ {Y} Y \rangle , $$

where $ Z _ {\mathfrak M } $ is the projection of $ Z \in \mathfrak G $ on $ \mathfrak M $ along $ \mathfrak H $.

The Nomizu operators can be expressed in terms of the Lie algebra $ \mathfrak G $ and the metric $ \langle , \rangle $ by the formula

$$ 2 \langle L _ {X} Y , Z \rangle = $$

$$ = \ \langle [ X , Y ] _ {\mathfrak M } , Z \rangle - < X , [ Y ,\ Z ] _ {\mathfrak M } > - \langle Y , [ X , Z ] _ {\mathfrak M } \rangle , $$

where $ X \in \mathfrak G $, $ Y , Z \in \mathfrak M $. It follows from the definition of the Nomizu operators that their action on $ G $- invariant fields differs only in sign from that of the covariant derivative at the point $ x _ {0} $. If the Riemannian space $ ( G / H , m) $ does not contain flat factors in the de Rham decomposition, then the linear Lie algebra generated by the Nomizu operators $ L _ {X} $, $ X \in \mathfrak G $, is the same as the holonomy algebra (cf. Holonomy group) of the space $ ( G / H , M) $ at $ x _ {0} $.

A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Suppose, to begin with, that $ M = G $ is a Lie group acting on itself by left translations. Let $ \gamma _ {t} $ be a left-invariant geodesic of the metric $ m $ on the Lie group $ G $ and let $ X _ {t} = \gamma ^ {-1} \dot \gamma _ {t} $ be the curve in the Lie algebra $ \mathfrak G $ corresponding to it (the velocity hodograph). The curve $ X _ {t} $ satisfies the hodograph equation

$$ \dot{X} _ {t} - L _ {X _ {t} } X _ {t} = \dot{X} _ {t} - ( \mathop{\rm ad} ^ {*} X _ {t} ) X _ {t} = 0 , $$

where $ \mathop{\rm ad} ^ {*} X $ is the operator dual to the adjoint representation $ \mathop{\rm ad} X $. The geodesic $ \gamma _ {t} $ can be recovered in terms of its velocity hodograph $ X _ {t} $ from the differential equation $ \dot \gamma _ {t} = \gamma _ {t} X _ {t} $( which is linear if the group $ G $ is linear) or from the functional relations

$$ \langle X _ {t} , ( \mathop{\rm Ad} \gamma ( t) ) Y \rangle = \textrm{ const } ,\ Y \in \mathfrak G , $$

giving the first integrals of this equation. Thus, the description of the geodesics of the metric $ m $ reduces to the integration of the hodograph equation, which sometimes can be completely integrated. For example, in the case when the metric $ m $ is also invariant with respect to right translations, the geodesics passing through the point $ e $ are the one-parameter subgroups of $ G $. Such a metric exists on any compact Lie group. In the case of an arbitrary homogeneous space $ M = G / H $ an invariant metric $ m $ on $ G / H $ can be "lifted" to a left-invariant metric $ \widetilde{m} $ on $ G $ for which the natural bundle $ G \rightarrow G / H $ of the Riemannian space $ ( G , \widetilde{m} ) $ over the Riemannian space $ ( G / H , m ) $ is a Riemannian bundle, that is, the length of tangent vectors orthogonal to the fibre remains unaltered under projection. For this it is sufficient to extend the metric $ \langle , \rangle $ to the entire algebra $ \mathfrak G $ by setting

$$ \langle \mathfrak H , \mathfrak M \rangle = 0 \ \textrm{ and } \ \langle X , Y \rangle = \ - \mathop{\rm Tr} L _ {X} L _ {Y} \ ( X , Y \in \mathfrak H ) , $$

and carrying it over by left translations to a metric $ \widetilde{m} $ on $ G $. The geodesics of $ ( G / H , \widetilde{m} ) $ are projections of geodesics of $ ( G , m ) $ that are orthogonal to the fibres.

Since the function $ X \rightarrow \langle X , X \rangle $ on $ \mathfrak G $ is always a first integral of the hodograph equation (the energy integral), the corresponding equation of the vector field on $ \mathfrak G $ is tangent to the spheres $ \langle X , X \rangle = \textrm{ const } $. This implies the completeness of the hodograph equation and therefore also the completeness of any invariant Riemannian metric on a homogeneous space. For a pseudo-Riemannian metric the completeness property does not hold, in general. On the other hand, any invariant pseudo-Riemannian metric on a compact homogeneous space is complete.

See also Symmetric space.

References

[1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2]	A.Z. Petrov, "New methods in general relativity theory" , Moscow (1966) (In Russian)
[3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
[4]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[5]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1984)
[6]	A. Lichnerowicz, "Geometry of groups of transformations" , Noordhoff (1977) (Translated from French)
[7]	A.L. Besse, "Einstein manifolds" , Springer (1987)

Comments

A de Rham decomposition (of the tangent space $ T _ {x} M $ at a point $ x $) is defined as follows. Let $ x \in M $, let $ T _ {x} M $ be the tangent space at $ x $ and let $ \Phi _ {x} $ be the holonomy group of the Riemannian connection at $ x $. The group $ \Phi _ {x} $ acts on $ T _ {x} M $. Let $ T _ {x} ^ {(} 0) M $ be the subspace of tangent vectors that are left invariant under $ \Phi _ {x} $. Let $ T _ {x} ^ \prime M $ be the orthogonal complement of $ T _ {x} ^ {(} 0) M $ in $ T _ {x} M $ and let $ T _ {x} ^ \prime M = \sum_{i=1} ^ {r} T _ {x} ^ {(} i) M $ be a decomposition of $ T _ {x} ^ \prime M $ into mutually-orthogonal invariant irreducible subspaces. The decomposition

$$ T _ {x} M = \sum_{i=0}^ { r } T _ {x} ^ {(i)} M $$

is called a de Rham decomposition or a canonical decomposition.

An irreducible Riemannian manifold is one for which the holonomy group $ \Phi _ {x} $ acts irreducibly on $ T _ {x} M $( so that there is only one factor in the Rham decomposition of $ T _ {x} M $).

The de Rham decomposition theorem says that a connected simply-connected complete Riemannian manifold $ M $ is isometric to a direct product $ M _ {0} \times M _ {1} \times \dots \times M _ {r} $ where $ M _ {0} $ is a Euclidean space (possibly of dimension zero) and where the $ M _ {1} \dots M _ {r} $ are all simply-connected complete irreducible Riemannian manifolds. Such a decomposition is unique up to the order of the factors, [a1], Sect. IV. 6.

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)