Malaysian Journal of Mathematical Sciences, April 2019, Vol. 13(S)
Special Issue: 3rd International Conference on Mathematical Sciences and Statistics (ICMSS2018)
Geometric Characterization of Totally Geodesic SODE Submanifolds
Ahangari, F.
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
Investigating the geometry of the tangent bundle \((TM,\pi, M)\) over a smooth manifold \(M\) is one of the most significant fields of modern differential geometry and has remarkable applications in various problems specifically in the theory of physical fields. This significance provides a constructive setting for the development of novel notions and geometric structures such as systems of second order differential equations (SODE), metric structures, semisprays and nonlinear connections. Accordingly analysis of above mentioned concepts can be considered as a powerful tool for the thorough investigation of the geometric properties of a tangent bundle. This paper is devoted to exhaustive geometric analysis of totally geodesic SODE submanifolds. Investigating the induced SODE structure on submanifolds is our main objective in this paper. Indeed, it is demonstrated that the metric which is obtained from the metrizability of a given semispray, plays a fundamental role in inducing SODE structure on submanifolds. Particularly, a necessary and sufficient condition for an SODE submanifold to be totally geodesic is presented.
Keywords: SODE, semispray, dynamical covariant derivative, metrizability, nonlinear connection, totally geodesic submanifolds
