General Deformations of Sprays on Finsler Manifolds

In this paper, we investigate the concept of general deformations of a spray S on a manifold M. We then focus on a specific case, which we call a projective-like deformation. This type of deformation extends the notion of projective deformation but, unlike projective deformation, it does not necessarily preserve geodesics. We derive an explicit formula for the Jacobi endomorphism under projective-like deformations and analyze the conditions under which it remains invariant. As applications, we consider (α,β)-metrics and spherically symmetric metrics. We find a necessary and sufficient condition for an (α,β)-metric and the Riemannian metric α to be projectively related. Additionally, we provide and examine several explicit examples.