Differential Geometric Approach to Foliated Structures.

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(a) We get the following result :Let (M,F,g) be a codimension-q bundle-like foliation on a closed Riemannian manifold of positive curvature. (1) If q is even, then F has a compact leaf. (2) If q is odd, then F has a leaf whose closure is a closed codimsnsion-(q-1) submanifold.As a corollary, we extend Berger's famous result :Any Killing vector field on a closed Riemannian manifold with positive sectional curvature admits a zero point or a closed orbit.(b) We study the dual 1-form to the mean curvature vector of a foliation. We give a characterization of such 1-forms for codimension-one foliations. We also have a simple characterization when the foliation is a bundle foliation, and when the dual 1-form is basic.(c) We get the following result :Let (M,F,g) be a codimension-1 minimal foliation on a complete Riemannian manifold of non-negative Ricci curvature. If the growth of F is not greater than 2, then F is totally geodesic. Further, (M,g) is locally a Riemannina product.As a byproduct, we get a simple proof of Mirand's result on minimal graphs, and foliated version of the result by Alencar and do Carmo on constant mean curvature hypersurfaces.