Hokkaido Mathematical Journal

INOGUCHI Jun-ichi, NAITOH Hiroo,

Grassmann geometry on the 3-dimensional unimodular Lie groups I.

Hokkaido Mathematical Journal, 38 (2009) pp.427-496

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Abstract

We study the Grassmann geometry of surfaces when the ambient space is a $3$-dimensional unimodular Lie group with left invariant metric, that is, it is one of the $3$-dimensional commutative Lie group, the $3$-dimensional Heisenberg group, the groups of rigid motions on the Euclidean or the Minkowski planes, the special unitary group $SU(2)$, and the special real linear group $SL(2,\mathbb R)$.

MSC(Primary)53B25
MSC(Secondary)53C40, 53C30
Uncontrolled KeywordsGrassmann geometry, unimodular Lie group, Heisenberg group, Euclidean plane, Minkowski plane, special unitary group, special linear group, totally geodesic surface, flat surface, minimal surface, surface of constant mean curvature