• Title/Summary/Keyword: Riemannian

Search Result 543, Processing Time 0.02 seconds

THE SYMMETRY OF spinℂ DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

  • HONG, KYUSIK;SUNG, CHANYOUNG
    • Journal of the Korean Mathematical Society
    • /
    • v.52 no.5
    • /
    • pp.1037-1049
    • /
    • 2015

  • It is well-known that the spectrum of a $spin^{\mathbb{C}}$ Dirac operator on a closed Riemannian $spin^{\mathbb{C}}$ manifold $M^{2k}$ of dimension 2k for $k{\in}{\mathbb{N}}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M^{2p}_1{\times}M^{2q+1}_2$ with a product $spin^{\mathbb{C}}$ structure for $p{\geq}1$, $q{\geq}0$, the spectrum of a $spin^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $spin^{\mathbb{C}}$ Dirac spectrum of $M^{2q+1}_2$ is symmetric or $(e^{{\frac{1}{2}}c_1(L_1)}{\hat{A}}(M_1))[M_1]=0$, where $L_1$ is the associated line bundle for the given $spin^{\mathbb{C}}$ structure of $M_1$.

  • https://doi.org/10.4134/JKMS.2015.52.5.1037 인용 PDF KSCI

INTRODUCTION OF T -HARMONIC MAPS

  • Mehran Aminian
    • The Pure and Applied Mathematics
    • /
    • v.30 no.2
    • /
    • pp.109-129
    • /
    • 2023

  • In this paper, we introduce a second order linear differential operator T□: C∞ (M) → C∞ (M) as a natural generalization of Cheng-Yau operator, [8], where T is a (1, 1)-tensor on Riemannian manifold (M, h), and then we show on compact Riemannian manifolds, divT = divTt, and if divT = 0, and f be a smooth function on M, the condition T□ f = 0 implies that f is constant. Hereafter, we introduce T-energy functionals and by deriving variations of these functionals, we define T-harmonic maps between Riemannian manifolds, which is a generalization of Lk-harmonic maps introduced in [3]. Also we have studied fT-harmonic maps for conformal immersions and as application of it, we consider fLk-harmonic hypersurfaces in space forms, and after that we classify complete fL1-harmonic surfaces, some fLk-harmonic isoparametric hypersurfaces, fLk-harmonic weakly convex hypersurfaces, and we show that there exists no compact fLk-harmonic hypersurface either in the Euclidean space or in the hyperbolic space or in the Euclidean hemisphere. As well, some properties and examples of these definitions are given.

  • https://doi.org/10.7468/jksmeb.2023.30.2.109 인용 PDF

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.

  • https://doi.org/10.4134/BKMS.b220452 인용 PDF