Isometry Groups of Riemannian Solvmanifolds

A simply connected solvable Lie group together with a left-invariant Riemannian metric is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds and <!-- MATH $(R' ,\,g' )$ --> may be isometric even when and are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization of a given solvmanifold, (ii) describing the embedding of in the full isometry group , and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of for arbitrary solvmanifolds .