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.
