HORIZONTAL LAPLACIAN ON TANGENT BUNDLE OF FINSLER MANIFOLD WITH g-NATURAL METRIC

2012 ◽  
Vol 09 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI ◽  
CHUNPING ZHONG
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Vector Bundles ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Order Differential Operator ◽  
Finsler Manifolds ◽  
Scalar And Vector Fields ◽  
Matsumoto Metric ◽  
Weitzenböck Formula

Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenböck formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.

Laplacians for the holomorphic tangent bundles with g-nature metrics on complex Finsler manifolds

2017 ◽  
Vol 28 (09) ◽  
pp. 1740011
Author(s):  
Hongjun Li ◽  
Chunhui Qiu ◽  
Weixia Zhu
Keyword(s):  
Laplace Operator ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Killing Vector Fields ◽  
Finsler Manifolds ◽  
Complex Finsler Manifold ◽  
Precise Relationship ◽  
Tangent Bundles

Let [Formula: see text] be a strongly pseudoconvex compact complex Finsler manifold. We first introduce a class of [Formula: see text]-nature metric [Formula: see text] for the slit holomorphic tangent bundle [Formula: see text] on [Formula: see text]. Then, we define the complex horizontal Laplacian [Formula: see text], and complex vertical Laplacian [Formula: see text], and obtain a precise relationship among [Formula: see text], [Formula: see text] and the Hodge–Laplace operator [Formula: see text] on [Formula: see text]. As an application, we discuss the holomorphic Killing vector fields associated to [Formula: see text].

Killing vector fields on compact Finsler manifolds

2016 ◽  
Vol 88 (1-2) ◽  
pp. 3-19
Author(s):  
JINLING LI ◽  
CHUNHUI QIU ◽  
TONGDE ZHONG
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Finsler Manifolds

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

2018 ◽  
Vol 62 (3) ◽  
pp. 623-641
Author(s):  
Bin Shen
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Vanishing Theorems ◽  
Complex Manifolds ◽  
Finsler Manifolds ◽  
Holomorphic Vector Fields ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Sections

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

Killing vector fields on compact Finsler manifolds

2016 ◽  
Vol 88 (1-2) ◽  
pp. 3-19
Author(s):  
JINLING LI ◽  
CHUNHUI QIU ◽  
TONGDE ZHONG
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Finsler Manifolds

scholarly journals A Geometric characterization of Finsler manifolds of constant curvatureK=1

2000 ◽  
Vol 23 (6) ◽  
pp. 399-407 ◽  
Author(s):  
A. Bejancu ◽  
H. R. Farran
Keyword(s):  
Vector Field ◽  
Constant Curvature ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Geometric Characterization ◽  
Killing Vector Field ◽  
Finsler Manifolds

We prove that a Finsler manifold𝔽mis of constant curvatureK=1if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundleIMof𝔽m.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

A Multi-Parameter Integrated Magnetometer Based on Combination of Scalar and Vector fields

2021 ◽  
pp. 1-1
Author(s):  
Jian Ge ◽  
Rui Wang ◽  
Haobin Dong ◽  
Huan Liu ◽  
Qianwei Zheng ◽  
...  
Keyword(s):  
Vector Fields ◽  
Scalar And Vector Fields
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