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DifferentialGeometry Tutorials


An Introduction to Homogeneous Spaces




Procedures Illustrated

Part A.  Algebraic Steps

Part B.  Constructing the homogeneous space

Part C.  An invariant metric on the homogenous space.



Let G be a Lie group action acting on a manifold M.  If the action is transitive (that is, if for any x, y in M there is g in G such that g*x = y), then M is called a homogeneous space (or, more precisely, a homogenous G space).  The isotropy subgroup for the action of G on M at a point x in M is H = {x in g such that g*x = x}. The infinitesimal generators for the action of G on M define a Lie algebra of vector fields on M which is isomorphic (assuming G acts effectively on M) to the Lie algebra g of the Lie group of G.  The infinitesimal isotropy subalgebra at x consists of those infinitesimal generators which vanish at x. This is a subalgebra h of  g which is isomorphic to the Lie algebra of H.


Two homogeneous G spaces M and N are said to be equivalent is there is a diffeomorphism phi : M -->N such that for any x in M and g in G, phi(g*x) = g*phi(x).


Let M be a homogenous G space with isotropy subgroup H at x0 and let G/H be the space of right cosets of H in G. Then the mapping phi which sends the coset gH in G/H to the point g*x0 in M is a bijective correspondence.  In fact, with respect to the natural manifold structure on G/H, the map phi defines a diffeomorphism from G/H to M so that the homogenous G spaces M and G/H are equivalent.


In this tutorial we start with a 4-dimensional solvable Lie algebra g and a 1 dimensional subalgebra h and construct a homogeneous space M whose Lie algebra of

infinitesimal generators is g and with infinitesimal isotropy subalgebra h.  Specifically we shall do the following:


1.  Use the DifferrentialGeometry Library to define a 4 dimensional Lie algebra g.

2.  Pick a 1 dimensional subalgebra h and find a reductive complement m to h in g.

3.  Find an h  invariant inner product < , > on m.

4.  Construct a (global) Lie group G whose Lie algebra is g.

5.  Construct the left and right invariant vector fields on G. Construct the Maurer Cartan forms on G.

6.  Construct the homogeneous space G -> G/H.

7.  Construct the action of G on M = G/H.

8.  Use the inner product on m to construct a G invariant metric on M = G/H.

9.  Define an orthonormal frame on M and calculate the curvature of the metric.


Procedures Illustrated

This is a comprehensive tutorial which illustrates a large number of commands from the DifferentialGeometry package and its subpackages.

DifferentialGeometry, LieAlgebras, Tensor, Library,GroupActions, Browse, ComplementaryBasis, Cotton tensor, DGsetup, DGinfo, GenerateSymmetricTensors, InfinitesimalTransformation, InvariantVectorsAndForms, ,LieDerivative, LieGroup, PushPullTensor, Query, References, Retrieve.

Part A.  Algebraic Steps


We first load in all the packages we shall need for this tutorial.







Our goal is to construct a 3 dimensional homogeneous space for a 4-dimensional Lie group. We begin by looking at the 4 dimensional Lie algebras available for our use.  All 4-dimensional Lie algebras have been classified and the results of these classifications are contained in the DifferentialGeometry Library.  The References command gives us a list of the articles and books whose results are in the DifferentialGeometry Library.


Doubrov, 1
         Classification of Subalgebras in the Exceptional Lie Algebra of Type G_2
         Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2008, No.3

Gong, 1
         Classification of Nilpotent Lie Algebras of Dimension 7( Over Algebraically Closed Fields and R)
         PhD. Thesis,  University of Waterloo (1998)

Gonzalez-Lopez, 1
         Lie algebras of vector fields in the real plane (with Kamran and Olver)
         Proc. London Math Soc. Vol 64 (1992), 339--368

Kamke, 1
          Chelsa Publ. Co. (1947)

Mubarakzyanov, 1
         Lie algebras of dimmensions 3, 4
         Izv. Vyssh. Uchebn. Zaved. Math 34(1963) 99

Mubarakzyanov, 2
         Lie algebras of dimension 5
         Izv. Vyssh. Uchebn. Zaved. Math 34(1963) 99

Mubarakzyanov, 3
         Lie algebras of dimension 6
         Izv. Vyssh. Uchebn. Zaved. Math 35(1963) 104

Olver, 1:
         Equivalence, Invariants and Symmetry, 472--473

Petrov, 1:
         Einstein Spaces

Stephani, 1:
         Exact Solutions to Einstein's Field Equations, 2nd Edition (with Kramer, Maccallum, Hoenselaers, Herlt)

Turkowski, 1:
         Low dimensional real Lie algebras
         JMP(29), 1990, 2139--2144

Turkowski, 2
         Solvable Lie Algebras of dimension six
         JMP(31), 1990, 1344--1350

Winternitz, 1:
         Invariants of real low dimensional Lie algebras, (with Patera, Sharp and Zassenhaus)
         JMP vol 17, No 6, June 1976, 966--994




The paper by Winternitz Invariants of real low dimensional algebras contains a convenient list of all Lie algebras of dimension  <= 5 which we will shall use here. The indices by which this these Lie algebras are labeling in the paper can be obtained using the Browse command.

Browse("Winternitz", 1);




Let us look specifically at the 4 dimensional Lie algebras.

Browse("Winternitz", 1,[[4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12]]);