• 대한수학회지
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• 제36권5호
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• pp.959-1008
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• 1999

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

• Korean Journal of Mathematics
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• 제32권3호
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• pp.467-484
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• 2024

In this paper, following the pullback approach to global Finsler geometry, we investigate a coordinate-free study of Shen square metric in a more general manner. Precisely, for a Finsler metric (M, L) admitting a concurrent π-vector field, we study some geometric objects associated with ${\widetilde{L}}(x, y)={\frac{(L+{\mathfrak{B}}^2)}L}$ in terms of the objects of L, where ${\mathfrak{B}}$ is the associated 1-form. For example, we find the geodesic spray, Barthel connection and Berwald connection of ${\widetilde{L}}(x,y)$. Moreover, we calculate the curvature of the Barthel connection of ${\tilde{L}}$. We characterize the non-degeneracy of the metric tensor of ${\widetilde{L}}(x,y)$.