Vol. 29, No. 3, 1969

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A special deformation of the metric with no negative sectional curvature of a Riemannian space

Grigorios Tsagas

Vol. 29 (1969), No. 3, 715–725

The main results of this paper can be stated as follows. Let M1, M2 be two big open submanifolds of the Riemannian manifolds (R12,h1) and (R22,h2), respectively. The submanifolds M1, M2 with the metrics h1∕M1 and h2∕M2, respectively, have positive constant sectional curvature. We have constructed a special I-parameter family of Riemannian metrics d(t) on M1 × M2 which is the deformation of the product metric h1∕M1 × h2∕M2 and it has strictly positive sectional curvature. In other words, we have proved that P M1 × M2 the derivative of the sectional curvature with respect to the parameter t for t = 0 and for any plane which is spanned by X (M1)p and Y (M2)p is strictly positive.

Mathematical Subject Classification
Primary: 53.70
Received: 16 July 1968
Published: 1 June 1969
Grigorios Tsagas