On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric

The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric. 1. Introduction The concept of Finsler spaces with -metric was introduced by Matsumoto [1]. It was later discussed by several authors such as Shibata and others [2–4]. It has plenty of applications in various fields such as physics, mechanics, seismology, biology, and ecology [5–8]. Matsumoto introduced a special type of -metric of the form , , and , which is slope-of-a-mountain metric and is known as Matsumoto metric [9]. This metric has enriched Finsler geometry and it has provided researchers an important tool to work with significantly in this field [7, 10]. A change of Finsler metric is called Randers change of metric. The notion of a Randers change was proposed by Matsumoto, named by Hashiguchi and Ichijyo [11] and studied in detail by Shibata [12]. A Randers change of Matsumoto metric is given by . Recently, Nagaraja and Kumar [13] studied the properties of a Finsler space with the Randers change of Matsumoto metric. Matsumoto [14] presented the theory of Finslerian hypersurface. The present authors (Gupta and Pandey [15, 16]) obtained certain geometrical properties of hypersurfaces of some special Finsler spaces. Singh and Kumari [17] discussed a hypersurface of a Finsler space with Matsumoto metric. In this paper, we consider an -dimensional Finsler space with the Randers change of Matsumoto metric and find certain geometrical properties of a hypersurface of the Finsler space with above metric. The paper is organized as follows. Section 2 consists of Preliminaries relevant to the subsequent sections. The induced Cartan connection for hypersurface of a Finsler space is defined in Section 3. Necessary and sufficient conditions under which the hypersurface of the above Finsler space is a hyperplane of first, second and third kind are obtained in Section 4. 2. Preliminaries Let be an -dimensional smooth manifold and let be an -dimensional Finsler space equipped with Randers change of Matsumoto metric function The derivative of above Randers change of Matsumoto metric with respect to and is given by where The normalized element of support is given by where . The angular metric tensor is given by where The fundamental metric tensor is given by where Moreover, the reciprocal tensor of is given by where The Cartan tensor is given by where Let be the components of Christoffel symbols of the associated Riemannian space and let be the covariant differentiation with respect to relative to this Christoffel symbols. We will