In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric [Formula: see text] and a one-form [Formula: see text]. They are called general [Formula: see text]-metrics. We have proven that, every Landsberg general [Formula: see text]-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general [Formula: see text]-metrics.

Let $(M_n,g)$ be a Riemannian manifold and $TM_n$ the total space of its tangent bundle. In this paper, we determine the infinitesimal fiber-preserving holomorphically projective (IFHP) transformations on $TM_n$ with respect to the Levi-Civita connection of the deformed complete lift metric $\tilde{G}_f=g^C+(fg)^V$, where $f$ is a nonzero differentiable function on $M_n$ and $g^C$ and $g^V$ are the complete lift and the vertical lift of $g$ on $TM_n$, respectively. Morevore, we prove that every IFHP transformation on $(TM_n,\tilde{G}_f)$ is reduced to an affine and induces an infinitesimal affine transformation on $(M_n,g)$.

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