Classification of five-dimensional generalized pointwise symmetric Riemannian spaces
A Riemannian manifold \((M,g)\) is said to be generalized pointwise symmetric if it admits a family \(\{s_x\}\), \(x \in M\), of isometries such that each \(x \in M\) is an isolated fixed point of the corresponding \(s_x\). The reviewer published in 1975 the explicit classification of all such (simply connected) manifolds up to dimension five under the assumption that the family \(\{s_x\}\) satisfies the so-called regularity condition \[ s_x \circ s_y = s_z \circ s_x,\quad \text{where } z = s_x (y). \] The last condition holds for every Riemannian symmetric space \((M,g)\), where \(\{s_x\}\) is the family of corresponding involutive symmetries. The present author in fact proves that, in dimension \(n = 5\), the existence of a ``general'' family \(\{s_x\}\) on \((M,g)\) implies the existence of a regular family \(\{s_x'\}\) and hence the classification follows. (For dimensions less than five the classification was published by the same author in 1988.) A short historical overview with references is also given in this article. One should remark that a geodesic sphere \(S^9\) of the complex projective space \(\mathbb{C} P^5\) (with the induced metric) admits a general family \(\{s_x\}\) but not any regular family \(\{s_x'\}\). (This point is not presented correctly in the paper.) The minimal dimension in which this phenomenon occurs is not known up to now, but the present paper implies that it must be one of the numbers 6, 7, 8, 9.
scientific article; zbMATH DE number 806271
scientific article