Study of Projective Curvature Tensor, 𝑾𝒋𝒌𝒉𝒊(𝒙,𝒙̇) in Bi-Recurrent Finsler Space, 𝟐𝑹−𝑭𝒏
Opondo, Mary Atieno
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The Weyl (W) projective curvature tensor, 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) has properties that have a wide range of applications in various fields but are still not easy to understand in a fundamental way since 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is a function of both position and direction. In this study three selected properties of the projective curvature tensor, 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) namely inheritance symmetry, collineation symmetry and decomposition are investigated in bi-recurrent Finsler space for purposes of new applications. Many authors have studied inheritance, collineation and decomposition properties for 𝐻,𝐾,𝑁,𝑅 and 𝑈 curvature tensors in recurrent Finsler spaces and in this thesis we have extended these studies to 𝑊 curvature tensor in bi-recurrent Finsler space which is still relatively under explored. The Finsler space is viewed as regular metric space and the three properties are described from the modern geometry view point using the relevant geometric and symbolic computation tools such as Lie derivatives, transvection, commutation and covariant differentiation in the sense of Berwald. The W-Curvature inheritance is defined by a Lie derivative (𝐿𝑣) proportional to the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) while the W-Curvature collineation is defined by vanishing Lie derivative (𝐿𝑣) of 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇). The results of the study show that every motion admitted in a bi-recurrent Finsler space is also a W-curvature inheritance if the space is isotropic otherwise it is a W-curvature collineation. The contra field and concurrent field are considered as special cases. The study reveals that both fields do not admit a W- curvature inheritance however they both admit a W- curvature collineation when the vector field (𝑉𝑖) of the infinitesimal transformation is orthogonal to the recurrence vector (𝐾𝑙). The geometrical properties of both inheritance and collineation symmetries have physical significance which make them useful in spacetime applications. The decomposition property of the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is also investigated for specified decomposition tensors denoted by Ψ(𝑥,𝑥̇) using different symbolic tensor computation algorithms. The study has established that tensor decomposition for the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is not unique but specific conditions discussed in this study introduce uniqueness into the decomposition algorithm. The study has also established that in both recurrent and bi-recurrent Finsler spaces the decomposition tensors have some properties similar to those of the original tensor and therefore decomposition can be used to compress tensors for further applications. The tensorial computation algorithms are presented in form of step by step equations and the principal results obtained have helped to identify some hidden components of the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇). The results of the study are summarized in form of theorems which have already been verified and can be used in various fields for theoretical investigations and practical applications of tensors.