a1 Department of Mathematics University of Ioannina Ioannina, Greece
a2 Department of Mathematics University of Southampton Southampton, England
Let N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.
(Received December 11 1980)
(Revised June 15 1981)
1980 Mathematics subject classification (Amer. Math. Soc.)