• Title/Summary/Keyword: isotropic submanifold

Search Result 3, Processing Time 0.017 seconds

CHARACTERIZATIONS OF SOME ISOMETRIC IMMERSIONS IN TERMS OF CERTAIN FRENET CURVES

  • Choi, Jin-Ho;Kim, Young-Ho;Tanabe, Hiromasa
    • Bulletin of the Korean Mathematical Society
    • /
    • v.47 no.6
    • /
    • pp.1285-1296
    • /
    • 2010
  • We give criterions for a submanifold to be an extrinsic sphere and to be a totally geodesic submanifold by observing some Frenet curves of order 2 on the submanifold. We also characterize constant isotropic immersions into arbitrary Riemannian manifolds in terms of Frenet curves of proper order 2 on submanifolds. As an application we obtain a characterization of Veronese embeddings of complex projective spaces into complex projective spaces.

CLASSIFICATION OF TWISTED PRODUCT LIGHTLIKE SUBMANIFOLDS

  • Sangeet Kumar;Megha Pruthi
    • Bulletin of the Korean Mathematical Society
    • /
    • v.60 no.4
    • /
    • pp.1003-1016
    • /
    • 2023
  • In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M×ϕMT, where M is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type MT×ϕM of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.