A Riemannian space with strictly positive sectional curvature

Grigorios Tsagas

Vol. 25 (1968), No. 2, 381–391

DOI: 10.2140/pjm.1968.25.381

Abstract

Let M₁ and M₂ be two Riemannian spaces¹ with Riemannian metrics d₁ and d₂ respectively whose sectional curvature is positive constant. We consider the product of the two Riemannian spaces M₁ × M₂, then the Riemannian space M₁ × M₂ has nonnegative sectional curvature with respect to the Riemannian metric d₁ × d₂ but not strictly positive sectional curvature.

We can construct a Riemannian metric on M₁ × M₂ which approaches the Riemannian metric d₁ × d₂ as closely as we wish and which has strictly positive sectional curvature.

Mathematical Subject Classification

Primary: 53.72

Milestones

Received: 14 March 1967

Published: 1 May 1968