The Existence of Soliton Metrics for Nilpotent Lie Groups

Details The Existence of Soliton Metrics for Nilpotent Lie Groups The Existence of Soliton Metrics for Nilpotent Lie Groups

2008-09-29

arxiv.org/abs/0809.5068v1

Mathematics

Differential Geometry

47 pages

Scientific paper

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N,g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac-Moody algebras to analyze the solution spaces for such linear systems. We use these methods to find infinitely many new examples of nilmanifolds with soliton metrics. We give a sufficient condition for a sum of soliton metric nilpotent Lie algebra structures to be soliton, and we use this criterion to show that soliton metrics exist on every naturally graded filiform metric Lie algebra.

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