Invariant metric
A Riemannian metric on a manifold
that does not change under any of the transformations of a given Lie group
of transformations. The group
itself is called a group of motions (isometries) of the metric
(or of the Riemannian space
).
A Lie group of transformations of a manifold
acting properly on
(that is, the mapping
,
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)
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of any point is a compact subgroup of
. If
itself is compact, then a
-invariant metric
can be constructed on
by averaging any metric
on
over
:
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where the integral is taken with respect to the Haar measure.
If is transitive,
can be identified with the space of cosets
of
with respect to the stabilizer
of a fixed point
, and in order that there exist a
-invariant metric on
it is necessary and sufficient that the linear isotropy group (see Isotropy representation) has compact closure in
(in particular, it is sufficient that
be compact). In this case the space
is reductive, that is, the Lie algebra
of
admits a decomposition
, where
is the subalgebra corresponding to
and
is a subspace that is invariant under
where
is the adjoint representation of
(cf. Adjoint representation of a Lie group). If
is identified with
, then any
-invariant metric
on
is obtained from some
-invariant Euclidean metric
on
in the following way:
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where is such that
.
The tensor fields associated with a -invariant metric (the curvature tensor, its covariant derivatives, etc.) are
-invariant fields. In the case of a homogeneous space
, their value at a point
can be expressed in terms of the Nomizu operator
, which is defined by the formula
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where is the velocity field of the one-parameter group of transformations
,
is the covariant differentiation operator of the Riemannian connection and
is the Lie derivative operator. In particular, the curvature operator
and the sectional curvature
in the direction given by the orthonormal basis
satisfy the following formulas:
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where is the projection of
on
along
.
The Nomizu operators can be expressed in terms of the Lie algebra and the metric
by the formula
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where ,
. It follows from the definition of the Nomizu operators that their action on
-invariant fields differs only in sign from that of the covariant derivative at the point
. If the Riemannian space
does not contain flat factors in the de Rham decomposition, then the linear Lie algebra generated by the Nomizu operators
,
, is the same as the holonomy algebra (cf. Holonomy group) of the space
at
.
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 is a Lie group acting on itself by left translations. Let
be a left-invariant geodesic of the metric
on the Lie group
and let
be the curve in the Lie algebra
corresponding to it (the velocity hodograph). The curve
satisfies the hodograph equation
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where is the operator dual to the adjoint representation
. The geodesic
can be recovered in terms of its velocity hodograph
from the differential equation
(which is linear if the group
is linear) or from the functional relations
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giving the first integrals of this equation. Thus, the description of the geodesics of the metric reduces to the integration of the hodograph equation, which sometimes can be completely integrated. For example, in the case when the metric
is also invariant with respect to right translations, the geodesics passing through the point
are the one-parameter subgroups of
. Such a metric exists on any compact Lie group. In the case of an arbitrary homogeneous space
an invariant metric
on
can be "lifted" to a left-invariant metric
on
for which the natural bundle
of the Riemannian space
over the Riemannian space
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
to the entire algebra
by setting
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and carrying it over by left translations to a metric on
. The geodesics of
are projections of geodesics of
that are orthogonal to the fibres.
Since the function on
is always a first integral of the hodograph equation (the energy integral), the corresponding equation of the vector field on
is tangent to the spheres
. 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 at a point
) is defined as follows. Let
, let
be the tangent space at
and let
be the holonomy group of the Riemannian connection at
. The group
acts on
. Let
be the subspace of tangent vectors that are left invariant under
. Let
be the orthogonal complement of
in
and let
be a decomposition of
into mutually-orthogonal invariant irreducible subspaces. The decomposition
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is called a de Rham decomposition or a canonical decomposition.
An irreducible Riemannian manifold is one for which the holonomy group acts irreducibly on
(so that there is only one factor in the Rham decomposition of
).
The de Rham decomposition theorem says that a connected simply-connected complete Riemannian manifold is isometric to a direct product
where
is a Euclidean space (possibly of dimension zero) and where the
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) |
Invariant metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_metric&oldid=47415