Invariant object

on a homogeneous space

A field of geometric quantities on a homogeneous space $ M = G / H $ of a Lie group $ G $ that does not change under any of the transformations of $ G $. A more rigorous definition of an invariant object is as follows. Let

$$ \pi : E \rightarrow M = G / H $$

be a locally trivial homogeneous fibration over a homogeneous space $ M = G / H $ of a Lie group $ G $, where $ G $ acts on $ E $ and $ \pi $ is equivariant under the action of $ G $ on $ E $ and $ M $. Then a section of $ \pi $ is called an invariant object (of type $ \pi $) on $ M $ if it is invariant under the action $ L ^ {E} $ of $ G $ in the space $ \Gamma ( E ) $ of sections of this fibration. The set of invariant objects of type $ \pi $ is in natural one-to-one correspondence with the set of $ H $- invariant elements of the fibre of the given fibration over the point $ m \in M $ corresponding to the coset $ e H $.

The most important and most studied special case is when $ \pi $ is a vector bundle. In this case, $ H $ acts linearly in the fibre over the point $ m $, and the invariant objects of type $ \pi $ are in one-to-one correspondence with the $ H $- invariant vectors of this fibre, which reduces their classification to a classification problem in the theory of invariants. For tensor bundles (associated with the tangent bundle) the problem of classifying the invariant objects reduces to finding the invariants of the linear isotropy group.

Invariant objects frequently arise in the following context. Let $ s $ be a field of geometric quantities (a geometric object) on a smooth manifold $ M $ and let $ \mathop{\rm Aut} ( s) $ be its group of symmetries, that is, the set of diffeomorphisms $ \phi $ of $ M $ such that $ \phi ^ {*} s = s $, where $ \phi ^ {*} $ is the transformation of $ s $ induced by $ \phi $. Suppose further that the group $ \mathop{\rm Aut} ( s) $ contains a subgroup $ G $ that acts transitively on $ M $ and is a Lie group. Then $ M $ can be identified with the homogeneous space $ G / H $, where $ H $ is the stationary subgroup (cf. Invariant subgroup) of an arbitrary point $ m \in M $, and the object $ s $ becomes an invariant object on the homogeneous space $ G / H $. Classical examples of invariant objects are an invariant Riemannian metric, an invariant complex structure, an invariant symplectic structure, an invariant contact structure, an invariant ordinary differential equation (in particular, a pulverization and a connection), and an invariant differential operator. A wide class of invariant objects admits a uniform description within the framework of the theory of $ G $- structures (cf. $ G $- structure).

Invariant objects arise naturally in various areas of mathematics and physics. For example, a linear differential equation with constant coefficients is an invariant object on Euclidean space regarded as a homogeneous space of a vector group. Eulerian motion of a rigid body arises as the geodesics of the left-invariant Riemannian metric on the group $ \mathop{\rm SO} ( 3) $. The homogeneity of Newton space and Minkowski space-time together with the Galilean principle of relativity lead to various invariant objects in Newtonian and relativistic physics, where the requirement of invariance often enables one virtually to determine the objects under consideration uniquely (the equation, the Lagrangian, etc.) (see [9]). The study of properties of invariant objects usually reduces easily to certain problems in linear algebra (often admitting a complete solution). This determines the important role of invariant objects as simple modelling examples clarifying a general situation. Often, invariant objects have a simpler structure than arbitrary objects of a given type. For example, in contrast to arbitrary Riemannian metrics, any invariant Riemannian metric on a homogeneous space is complete (as is any invariant pseudo-Riemannian metric on a compact homogeneous space), and any self-intersecting geodesic of it is closed.

Progress has been made on questions of the classification of invariant objects for a small number of the classical invariant tensor objects. Most complete results have been obtained for homogeneous spaces of compact Lie groups.

Of great interest from various points of view is the study of invariant objects on non-homogeneous $ G $- spaces, that is, geometric objects that are invariant under a given intransitive Lie group $ G $ of transformations of a manifold $ M $. Here, in the case of a compact Lie group, in order to construct invariant objects one often uses the method of averaging over the group (for example, the theorem on the existence of a $ G $- invariant Riemannian metric). A more refined method (applicable to a broader class of Lie groups of transformations, the so-called properly-acting transformation groups) is based on the existence of a slice, the existence of which implies the "almost local triviality" of the fibration of the manifold $ M $ on the orbits of the group $ G $.

An important generalization of the notion of invariant object is that of a covariant object. Suppose that an additional structure $ t _ {m} $ is given on the fibres $ F _ {m} $ of the fibration $ \pi $, smoothly depending on the point $ m \in M $( for example, a vector space structure), and let $ A _ {m} = \mathop{\rm Aut} ( t _ {m} ) $ be the group of automorphisms of the structure $ t _ {m} $ on the fibre $ F _ {m} $. The set of sections $ k : m \rightarrow S _ {m} \in A _ {m} $ of the fibration $ \psi : A = \cup _ {m \in M } A _ {m} \rightarrow M $ forms a group of automorphisms of $ \pi $, called the gauge group. Let $ K $ be a subgroup of it. Then a section $ s $ of $ \pi $ is called a $ K $- covariant object of type $ \pi $ on $ M = G / H $ if $ ( L _ {g} ^ {E} s ) ( m ) = k _ {g} ( m ) s ( m ) $ for all $ g \in G $, $ m \in M $, where $ g \mapsto k _ {g} $ is a homomorphism $ G \rightarrow K $. The most important particular case is obtained if $ \pi : E \rightarrow M = G / H $ is a vector bundle, $ K $ is the group of positive functions on $ M $, regarded as a group of automorphisms of $ \pi $, and $ k e = k ( m ) e $, where $ k \in K $, $ m = \pi ( e ) $, $ e \in E $. In this case a $ K $- covariant object is also called a conformally-invariant object, and the section of the corresponding bundle determined by it is an invariant object.

References

[1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
[2]	A. Lichnerowicz, "Geometry of groups of transformations" , Noordhoff (1977) (Translated from French)
[3]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1984)
[4]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[5]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[6]	B.P. Komrakov, "Differential-geometric structures and homogeneous spaces" , Minsk (1977) pp. Chapt. 1 (In Russian)
[7]	Itogi Nauk. Algebra Topol. Geom. 1962 (1964)
[8]	Itogi Nauk. Algebra Topol. Geom. 1965 (1967)
[9]	J.M. Lévy-Leblond, "Group-theoretical foundations of classical mechanics: The Lagrangian gauge problem" Commun. Math. Phys. , 12 : 1 (1969) pp. 64–79