5.2Mathematical Interlude: Isometries and the Geometry of Coset Manifolds

Suppose now that the considered diffeomorphism is infinitesimally close to the identity:

The condition for this diffeomorphism to be an isometry, is a differential equation for the components of the vector field k = kμ∂μ which immediately follows from (5.2.2):

Hence given a metric one can investigate the nature of its isometries by trying to solve the linear homogeneous equations (5.2.4) determining its general integral. The important point is that, if we have two Killing vectors k and w also their commutator [k, w] will be a Killing vector. This follows from the fact that the product of two finite isometries is also an isometry. Hence Killing vector fields form a finite dimensional Lie algebra Giso and one can turn the question around. Rather then calculating the isometries of a given metric one can address the problem of constructing (pseudo-)Riemannian manifolds that have a prescribed isometry algebra. Due to the well established classification of semi-simple Lie algebras this becomes a very fruitful point of view.

In particular, also in view of the Cosmological Principle, one is interested in homogeneous spaces, namely in (pseudo-)Riemannian manifolds where each point of the manifold can be reached from a reference one by the action of an isometry.

Homogeneous spaces are identified with coset manifolds, whose differential geometry can be thoroughly described and calculated in pure Lie algebra terms.

5.2.2 Coset Manifolds

Coset manifolds are a natural generalization of group manifolds and play a very important, ubiquitous, role both in Mathematics and in Physics.

In group-theory (irrespectively whether the group G is finite or infinite, continuous or discrete) we have the concept of coset space G/H which is just the set of equivalence classes of elements g G, where the equivalence is defined by right multiplication with elements h H G of a subgroup:

Namely two group elements are equivalent if and only if they can be mapped into each other by means of some element of the subgroup. The equivalence classes, which constitute the elements of G/H are usually denoted gH, where g is any representative of the class, namely any one of the equivalent G-group elements the class is composed of. The definition we have just provided by means of right multiplication can be obviously replaced by an analogous one based on left-multiplication. In this case we construct the coset H\G composed of right lateral classes Hg while gH

are named the left lateral classes. For non-Abelian groups G and generic subgroups H the left G/H and right H\G coset spaces have different not coinciding elements. Working with one or with the other definition is just a matter of conventions. We choose to work with left classes.

Coset manifolds arise in the context of Lie group theory when G is a Lie group and H is a Lie subgroup thereof. In that case the set of lateral classes gH can be endowed with a manifold structure inherited from the manifold structure of the parent group G. Furthermore on G/H we can construct invariant metrics such that all elements of the original group G are isometries of the constructed metric. As we show below, the curvature tensor of invariant metrics on coset manifolds can be constructed in purely algebraic terms starting from the structure constants of the G Lie algebra, by-passing all analytic differential calculations.

The reason why coset manifolds are relevant to Cosmology is encoded in the concept of homogeneity, that is one of the two pillars of the Cosmological Principle. Indeed coset manifolds are easily identified with homogeneous spaces which we presently define.

Definition 5.2.1 A Riemannian or pseudo-Riemannian manifold Mg is said to be homogeneous if it admits as an isometry the transitive action of a group G. A group acts transitively if any point of the manifold can be reached from any other by means of the group action.

A notable and very common example of such homogeneous manifolds is pro-

vided by the spheres Sn and by their non-compact generalizations, the pseudospheres H(n± +1−m,m). Let xI denote the Cartesian coordinates in Rn+1 and let:

be the coefficient of a non-degenerate quadratic form with signature (n + 1 − m, m):

We obtain a pseudo-sphere H(n± +1−m,m) by defining the algebraic locus:

which is a manifold of dimension n. The spheres Sn correspond to the particular case Hn++1,0 where the quadratic form is positive definite and the sign in the right hand side of (5.2.8) is positive. Obviously with a positive definite quadratic form this is the only possibility.

All these algebraic loci are invariant under the transitive action of the group SO(n + 1, n + 1 − m) realized by matrix multiplication on the vector x since:

namely the group maps solutions of the constraint (5.2.8) into solutions of the same and, furthermore, all solutions can be generated starting from a standard reference vector:

the line separating the first n + 1 − m entries from the last m. Equation (5.2.10) guarantees that the locus is invariant under the action G, while (5.2.11) states that G is transitive.

Definition 5.2.2 In a homogeneous space Mg , the subgroup Hp G which leaves a point p Mg fixed ( h Hp , hp = p) is named the isotropy subgroup of the point. Because of the transitive action of G, any other point p = gp has an isotropy subgroup Hp = gHpg−1 which is conjugate to Hp and therefore isomorphic to it.

It follows that, up to conjugation, the isotropy group of a homogeneous manifold Mg is unique and corresponds to an intrinsic property of such a space. It suffices to calculate the isotropy group H0 of a conventional properly chosen reference point p0: all other isotropy groups will immediately follow. For brevity H0 will be just

renamed H.

In our example of the spaces H(n± +1−m,m) the isotropy group is immediately derived by looking at the form of the vectors x±0 : all elements of G which rotate the vanishing entries of these vectors among themselves are clearly elements of the isotropy group. Hence we find:

It is natural to label any point p of a homogeneous space by the parameters describing the G-group element which carries a conventional point p0 into p. These parameters, however, are redundant: because of the H-isotropy there are infinitely many ways to reach p from p0. Indeed, if g does that job, any other element of the lateral class gH does the same. It follows by this simple discussion that the homogeneous manifold Mg can be identified with the coset manifold G/H defined by the transitive group G divided by the isotropy group H.