This article uses the Berwald connection exclusively, together with its two curvatures, to cut an efficient path across the landscape of Finsler geometry. Its goal is to initiate differential geometers into two key research areas in the field: the search for unblemished "unicorns" and the study of Ricci flow. The exposition is almost self-contained.

$(\alpha, \beta)$-metrics form a rich class of computable Finsler metrics. Many $(\alpha, \beta)$-metrics with special curvature properties have been found and discussed. They play an important role in Finsler geometry. In this article, we introduce the recent developments in the study of $(\alpha, \beta)$-metrics.

We study a connectedness principle in positively curved Finsler manifolds. Some results from positively curved Riemannian manifolds are generalized to Finsler spaces, and we also emphasize differences between the Riemann and Finsler settings.

This paper study the Gauss-Bonnet theorem for Finsler surfaces with smooth boundary. This is a natural generalization of the Gauss-Bonnet theorem for Riemannian surfaces with smooth boundary as well as an extension of the Gauss-Bonnet theorem for boundaryless Finsler surfaces. The paper starts with an introduction in the Finsler geometry of surfaces with emphasis on the Berwald and Landsberg surfaces.

Linear and metrical connections of a Riemannian space, whose indicatrices are ellipsoids, are established in the tangent bundle. Indicatrices of Finsler spaces are smooth, starshaped and convex hypersurfaces. They do not transform, in general, into each other by linear transformations, and thus they do not admit linear metrical connections in the tangent bundle. This necessitates the introduction of line-elements yielding the dependence of the geometric objects not only of points $x$ but also of the direction $y$. Therefore, the apparatus (connections, covariant derivatives, curvatures, etc.) of Finsler geometry becomes inevitably a little more complicated.

Nevertheless there are a number of problems which need no line-elements. Such are those, which concern the metric only (arc length, area, angle, geodesics, etc.) and also the investigation of those important special Finsler spaces, which allow linear metrical connections in the tangent bundle.

In this paper we want to present results which use the tangent bundle $TM$ only, and do not need $TTM$ or $VTM$ or line-elements. These investigations often admit direct geometrical considerations. Longer proofs are only sketched or omitted.

This paper is dedicated to the memory of Makoto Matsumoto, Finsler teacher and friend. Prof. Matsumoto augmented work of Darboux and proved that any 2-spray is projectively a geodesic spray of a Finsler manifold, [12]. This result has a refinement for the case of constant coefficient sprays, all of which are projectively equivalent to straight lines, [5].

In the present paper, we classify 2-sprays whose coefficients are linear in $x^1$, $x^2$, the adapted coordinates, by a perturbation technique. We also study the Feynman-Kac solutions to the corresponding Finslerian diffusions. The results herein arose from applications, especially [3], [4], [7], [10], [14].

The computations in this work have been performed by the computer package Finsler [1], [13].

Any path geometry, or projective equivalence class of sprays, on an $n$-dimensional manifold $M$ is naturally associated with an almost Grassmann structure on a $2n$-dimensional fibre bundle over that manifold. The almost Grassmann structure has special properties when the sprays are isotropic, and when they are geodesic for some Finsler function.

In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an outline of our main tools, i.e., the pull-back bundle formalism, we give an overview of Ehresmann connections and covariant derivatives in the pull-back bundle of a tangent bundle over itself. Then we define and characterize some special classes of generalized metrics. By a generalized metric we shall mean a pseudo-Riemannian metric tensor in our pull-back bundle. The main new results are contained in Section 5. We shall say, informally, that a metric covariant derivative is 'good' if it is related in a natural way to an Ehresmann connection determined by the metric alone. We shall find a family of good metric derivatives for the so-called weakly normal Moór–Vanstone metrics and a distinguished good metric derivative for a certain class of Miron metrics.

In Part I of this paper the dynamical systems of the Lagrangian mechanical system $\Sigma_L = (M, L(x,y),\ F_e(x,y))$ are defined and investigated. In Theorem 3.1 we prove the existence of a canonical dynamical system on the phase space whose integral curves are given by the Lagrange equations of $\Sigma_L$. The particular case of Finslerian mechanical systems is considered. The geometry of $\Sigma_L$ on TM is also described. Part I is a survey of the author's papers [18] [22] [23].

In the Part II for the first time the same problems for the Hamiltonian mechanical systems $\Sigma_H = (M, H(x,p),\ F_e(x,p))$ are studied. In Theorem 10.1, we prove the existence of a canonical dynamical system $\xi$ on the momenta space, whose integral curves are given by the Hamilton equations of $\Sigma_H$. As a particular case the Cartan mechanical systems are examined.

In this paper, we shall discuss the theory of connection in complex Finsler geometry, i.e., the Chern-Finsler connection $\nabla$ and its applications. In particular, we shall investigate (1) the ampleness of holomorphic vector bundles over a compact complex manifold which is based on the study due to [Ko1], (2) some special class of complex Finsler metrics and its characterization in terms of torsion and curvature of $\nabla$, and in the last section, (3) the characterization of Finsler-Kähler manifolds in terms of the Cartan connection $D$ which is naturally induced on the real tangent bundle from $\nabla$.