These are notes from Cheeger-Ebin. A Lie group is a manifold for which is smooth and a Lie algebra is a vector space with a pairing satisfying linearity, anticommutativity and the Jacobi identity . Vector fields on manifolds form an infinite dimensional Lie group with commutators satisfying the above axioms. The tangent space of a Lie group at a point is isomorphic to the left invariant vector fields on it and therefore has the structure of a Lie algebra. Any finite dimensional Lie algebra can be shown to be the Lie algebra of some Lie group. A subalgebra of a Lie algebra considered as the left invariant vector fields on a Lie group defines an involutive (meaning the brackets stay within the subspace) distribution so the Frobenius theorem guarantees a submanifold and the maximal submanifold through the identity is a subgroup. Conversely a subgroup of a Lie group then the tangent space is a Lie subalgebra which is a normal subalgebra if and only if it is an ideal. Since any one dimensional subalgebra is involutive, the corresponding subgroup is a one-parameter subgroup. A compact admits a bi-invariant metric.

If is a left-invariant metric:

(a)

(b)

(c)

Ok, so the homogeneous spaces are those which occur as for some subgroup and everyone’s favourite homogeneous manifolds are spheres and hyperbolic spaces which have constant curvatures. The map is a riemannian submersion defined by the property that is an isometry.

An interesting result is that if is a riemannian submersion and and is a horizontal lift then is a geodesic if and only if is.

The most interesting homogeneous space is probably the upper half complex plane with consisting of matrices with determinant 1 by Moebius transformations . Grassmannians , projective spaces are homogeneous, etc. The entire Thurston Geometrization program is to decompose compact three dimensional manifolds into homogeneous pieces. Yau likes to emphasize the search for special metrics like Einstein metrics . Most known Einstein metrics with are homogeneous.