The moduli space of left-invariant metrics on six-dimensional characteristically solvable nilmanifolds

Isolda Cardoso; Ana Cosgaya; Silvio Reggiani

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arXiv · 2404.19177

The moduli space of left-invariant metrics on six-dimensional characteristically solvable nilmanifolds

Isolda Cardoso·Ana Cosgaya·Silvio Reggiani

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Abstract

A real Lie algebra is said to be characteristically solvable if its derivation algebra is solvable. We explicitly determine the moduli space of left-invariant metrics, up to isometric automorphism, for $6$-dimensional nilmanifolds whose associated Lie algebra is characteristically solvable of triangular type. We also compute the corresponding full isometry groups. For each left-invariant metric on these nilmanifolds we compute the index and distribution of symmetry. In particular, we find the first known examples of Lie groups which do not admit a left-invariant metric with positive index of symmetry. As an application we study the index of symmetry of nilsoliton metrics. We prove that nilsoliton metrics detect the existence of left-invariant metrics with positive index of symmetry.

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Isolda Cardoso, Ana Cosgaya, Silvio Reggiani. 2024-04-30. The moduli space of left-invariant metrics on six-dimensional characteristically solvable nilmanifolds. https://arxiv.org/abs/2404.19177

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