• Title, Summary, Keyword: concircular vector field

Search Result 5, Processing Time 0.036 seconds

SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

  • CHEN, BANG-YEN
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.5
    • /
    • pp.1535-1547
    • /
    • 2015
  • A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

RIEMANNIAN SUBMANIFOLDS WITH CONCIRCULAR CANONICAL FIELD

  • Chen, Bang-Yen;Wei, Shihshu Walter
    • Bulletin of the Korean Mathematical Society
    • /
    • v.56 no.6
    • /
    • pp.1525-1537
    • /
    • 2019
  • Let ${\tilde{M}}$ be a Riemannian manifold equipped with a concircular vector field ${\tilde{X}}$ and M a submanifold (with its induced metric) of ${\tilde{M}}$. Denote by X the restriction of ${\tilde{X}}$ on M and by $X^T$ the tangential component of X, called the canonical field of M. In this article we study submanifolds of ${\tilde{M}}$ whose canonical field $X^T$ is also concircular. Several characterizations and classification results in this respect are obtained.

ON 3-DIMENSIONAL LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS

  • Chaubey, Sudhakar Kumar;Shaikh, Absos Ali
    • Communications of the Korean Mathematical Society
    • /
    • v.34 no.1
    • /
    • pp.303-319
    • /
    • 2019
  • The aim of the present paper is to study the Eisenhart problems of finding the properties of second order parallel tensors (symmetric and skew-symmetric) on a 3-dimensional LCS-manifold. We also investigate the properties of Ricci solitons, Ricci semisymmetric, locally ${\phi}$-symmetric, ${\eta}$-parallel Ricci tensor and a non-null concircular vector field on $(LCS)_3$-manifolds.

EUCLIDEAN SUBMANIFOLDS WITH CONFORMAL CANONICAL VECTOR FIELD

  • Chen, Bang-Yen;Deshmukh, Sharief
    • Bulletin of the Korean Mathematical Society
    • /
    • v.55 no.6
    • /
    • pp.1823-1834
    • /
    • 2018
  • The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component $x^T$ of the position vector field is the most natural vector field tangent to the Euclidean submanifold M. We simply call the vector field $x^T$ the canonical vector field of the Euclidean submanifold M. In earlier articles [4,5,9,11,12], we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.

Some Results on Null Hypersurfaces in (LCS)-manifolds

  • Ssekajja, Samuel
    • Kyungpook Mathematical Journal
    • /
    • v.59 no.4
    • /
    • pp.783-795
    • /
    • 2019
  • We prove that a Lorentzian concircular structure (LCS)-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there are no totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold.