THEOREM 3. Suppose n + 1 = 3. Let F and W be as in the original hypotheses with dW = 0 but assume the sectional curvatures of W lie between c x and c 2. Then we have 2c\ < Kf < 2c 2. DEFINITION OF GAUSSIAN CURVATURE. We define, for a Riemannian manifold I = l n , the function K: I-> R in two cases (which overlap): Case (i). n is even. In this case a local orthonormal frame on / gives rise to a matrix of curvature 2-forms, £2 = (fij.) defined locally. The Pfaffians of the local £1 agree on overlaps and so define a global «-form Pf(£2) on /. Letting v denote the canonical volume form on / we set 2 n/2 • (n/2)\ (see [3, vol. V, pp. 417-420]).
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