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)