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DifferentialGeometry Tutorials

An Introduction to Homogeneous Spaces

 Overview 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.

Part A.  Algebraic Steps

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

 > with(DifferentialGeometry):
 > with(LieAlgebras):
 > with(Tensor):
 > with(GroupActions):
 > with(Library):

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.

 > References(verbose);
 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           Differentialgleichungen           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
 $\left[\left[{"Doubrov"}{,}{1}\right]{,}\left[{"Gong"}{,}{1}\right]{,}\left[{"Gonzalez-Lopez"}{,}{1}\right]{,}\left[{"HawkingEllis"}{,}{1}\right]{,}\left[{"Kamke"}{,}{1}\right]{,}\left[{"Morozov"}{,}{1}\right]{,}\left[{"Mubarakyzanov"}{,}{1}\right]{,}\left[{"Mubarakyzanov"}{,}{2}\right]{,}\left[{"Mubarakyzanov"}{,}{3}\right]{,}\left[{"Olver"}{,}{1}\right]{,}\left[{"Petrov"}{,}{1}\right]{,}\left[{"Stephani"}{,}{1}\right]{,}\left[{"Turkowski"}{,}{1}\right]{,}\left[{"Turkowski"}{,}{2}\right]{,}\left[{"USU"}{,}{2}\right]{,}\left[{"Winternitz"}{,}{1}\right]\right]$ (3.1)

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);
 $\left[\left[{3}{,}{0}\right]{,}\left[{3}{,}{1}\right]{,}\left[{3}{,}{2}\right]{,}\left[{3}{,}{3}\right]{,}\left[{3}{,}{4}\right]{,}\left[{3}{,}{5}\right]{,}\left[{3}{,}{6}\right]{,}\left[{3}{,}{7}\right]{,}\left[{3}{,}{8}\right]{,}\left[{3}{,}{9}\right]{,}\left[{4}{,}{0}\right]{,}\left[{4}{,}{1}\right]{,}\left[{4}{,}{2}\right]{,}\left[{4}{,}{3}\right]{,}\left[{4}{,}{4}\right]{,}\left[{4}{,}{5}\right]{,}\left[{4}{,}{6}\right]{,}\left[{4}{,}{7}\right]{,}\left[{4}{,}{8}\right]{,}\left[{4}{,}{9}\right]{,}\left[{4}{,}{10}\right]{,}\left[{4}{,}{11}\right]{,}\left[{4}{,}{12}\right]{,}\left[{5}{,}{0}\right]{,}\left[{5}{,}{1}\right]{,}\left[{5}{,}{2}\right]{,}\left[{5}{,}{3}\right]{,}\left[{5}{,}{4}\right]{,}\left[{5}{,}{5}\right]{,}\left[{5}{,}{6}\right]{,}\left[{5}{,}{7}\right]{,}\left[{5}{,}{8}\right]{,}\left[{5}{,}{9}\right]{,}\left[{5}{,}{10}\right]{,}\left[{5}{,}{11}\right]{,}\left[{5}{,}{12}\right]{,}\left[{5}{,}{13}\right]{,}\left[{5}{,}{14}\right]{,}\left[{5}{,}{15}\right]{,}\left[{5}{,}{16}\right]{,}\left[{5}{,}{17}\right]{,}\left[{5}{,}{18}\right]{,}\left[{5}{,}{19}\right]{,}\left[{5}{,}{20}\right]{,}\left[{5}{,}{21}\right]{,}\left[{5}{,}{22}\right]{,}\left[{5}{,}{23}\right]{,}\left[{5}{,}{24}\right]{,}\left[{5}{,}{25}\right]{,}\left[{5}{,}{26}\right]{,}\left[{5}{,}{27}\right]{,}\left[{5}{,}{28}\right]{,}\left[{5}{,}{29}\right]{,}\left[{5}{,}{30}\right]{,}\left[{5}{,}{31}\right]{,}\left[{5}{,}{32}\right]{,}\left[{5}{,}{33}\right]{,}\left[{5}{,}{34}\right]{,}\left[{5}{,}{35}\right]{,}\left[{5}{,}{36}\right]{,}\left[{5}{,}{37}\right]{,}\left[{5}{,}{38}\right]{,}\left[{5}{,}{39}\right]{,}\left[{5}{,}{40}\right]{,}\left[{6}{,}{1}\right]{,}\left[{6}{,}{2}\right]{,}\left[{6}{,}{3}\right]{,}\left[{6}{,}{4}\right]{,}\left[{6}{,}{5}\right]{,}\left[{6}{,}{6}\right]{,}\left[{6}{,}{7}\right]{,}\left[{6}{,}{8}\right]{,}\left[{6}{,}{9}\right]{,}\left[{6}{,}{10}\right]{,}\left[{6}{,}{11}\right]{,}\left[{6}{,}{12}\right]{,}\left[{6}{,}{13}\right]{,}\left[{6}{,}{14}\right]{,}\left[{6}{,}{15}\right]{,}\left[{6}{,}{16}\right]{,}\left[{6}{,}{17}\right]{,}\left[{6}{,}{18}\right]{,}\left[{6}{,}{19}\right]{,}\left[{6}{,}{20}\right]{,}\left[{6}{,}{21}\right]{,}\left[{6}{,}{22}\right]\right]$ (3.2)

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]]);
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{1}\right]$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{2}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{a}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{3}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{4}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{5}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{a}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{b}{}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{6}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{a}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{b}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{b}{}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{7}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{8}\right]$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{9}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}\left({b}{+}{1}\right){}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{b}{}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{10}\right]$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{11}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{a}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{a}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{a}{}{\mathrm{e3}}\right]$
 ${\mathrm{___________________}}$
 ${"Winternitz"}{,}{1}{,}\left[{4}{,}{12}\right]$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}\right]$
 ${\mathrm{___________________}}$ (3.3)