Some Contributions to Projective and Conformal Transformations of Finsler space with and#945; and#946; metrics

Abstract

The research work on Finsler Geometry was in fact initiated by Riemann, sixty years earlier than Finsler. Riemann introduced the idea of manifolds, generalised metric, and provided a demonstration of Finsler metric in his monumental lecture. However, he said that the calculations of such type of examples are difficult. Mathematicians had not given this area much thought. On the other hand, Finsler was the one who first proposed the idea of it. The current research work exemplifies to extend the results on Finsler space with certain (and#945;, and#946;)-metric. Based on the features the thesis is divided into SIX chapters. Chapter 1 begins with the introduction which deals with the historical background of Finsler geometry, brief introduction to Finsler space, important literature reviews, development of Finsler Geometry and methods that have been used as an important tool. The Second chapter emphasises on projective relation between generalized (and#945;, and#946;)- metric with certain (and#945;, and#946;)-metric. We have shown that two important pairs of (and#945;, and#946;)- metrics, namely, generalized (and#945;, and#946;)-metric F = and#956;1and#945; + and#956;2and#946; + and#956;3 and#946;2 and#945; with Kropina metric and generalized (and#945;, and#946;)-metric with Randers metric, have same Douglas tensors if and only if both are Douglas metrics. Furthermore, we study the projective relation between two important (and#945;, and#946;)-metrics with dimension n and#8805; 0. Also, locally projective flat Finsler space with certain (and#945;, and#946;)-metric. The present study engages to investigate the projectively flat special (and#945;, and#946;)-metric and the generalised first approximate Matsumoto (and#945;, and#946;)-metric. Further, we concluded that and#945; is locally Projectively flat and and#946; is parallel with respect to and#945; for both the metrics. Finally, we have obtained necessary and sufficient conditions for the aforementioned metrics to be locally projectively flat.