On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold

Let M n be an n -dimensional smooth compact Riemannian manifold. By a theorem of Nash, we can think of it as an isometrically immersed submanifold in some higher dimensional Euclidean space ℝ n + m . Viewing in this way we can compare the intrinsic geometry of M to its extrinsic geometry. Classically, the Gauss equation where K(X,Y) denotes the sectional curvature in M corresponding to the plane spanned by the two orthonormal vectors X, Y and B denotes the second fundamental form gives one of the most important relations between the intrinsic and extrinsic geometries of M . In this note we shall prove the following.