Applications to Differential Geometry: The Nilmanifolds

Abstract

Of course, Lie algebras arise in a natural way in the study of transformation groups in differential geometry. But, nilpotent Lie algebras play an essential role in this field: they permit the construction of many examples of concrete differential manifolds and, more precisely, compact differential manifolds. These manifolds are called nilmanifolds and their differential calculus is, usually, nothing more than a linear calculus on the corresponding nilpotent Lie algebra. For example, the classical algebraic and topological invariants of a nilmanifold, such as the De Rham cohomological classes, are entirely defined in the Lie algebra. This explains why the exceptional examples of affine manifolds, of symplectic compact non-kahlerian manifolds, of naturally reductive Riemaniann manifolds, and the counter examples of Lichnerowicz’s conjecture on the harmonic spaces are described as nilmanifolds.