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In this paper, we are concerned with hypersurfaces in (n + 1) dimensional lightlike cone. We will give the fundamental theories and some properties of such hypersurfaces and characterize some of them. Finally, using the moving frame method, we calculate the first and the second variation formulas of the area integral for the hypersurface in lightlike cone.

We study transverse conformal Killing forms on foliations and prove a GallotMeyer theorem for foliations. Moreover, we show that on a foliation with C-positive normal curvature, if there is a closed basic 1-form φ such that B φ = qCφ, then the foliation is transversally isometric to the quotient of a q-sphere.

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Let (M, g M , F ) be a closed, connected Riemannian manifold with a foliation F of codimension q and a bundle-like metric g M . We study the relationship between several infinitesimal automorphisms. Moreover under the some curvature condition, if M admits a transversal conformal field, then F is transversally isometric to the action of a finite subgroup of O(q) acting on the q-sphere of constant curvature.