What is the difference/relation between Lorentzian and Finsler geometry? I'm kind of lost among many similar concepts. What is the difference/relation among Lorentzian manifold, Finsler manifold, Minkowsky manifold (and I just came across such a  Randers space, related to Minkowsky space... oh, my!). Can anybody there put it in an easy and concise way for me?
Many thanks in advance!
 A: All these spaces are modifications/generalizations of Riemannian Manifolds, which briefly put is a smooth manifold endowed with a smooth assignment of inner products (i.e., symmetric positive definite bilinear forms) to all of its tangent spaces:
\begin{align}
M\ni x\mapsto \left(g_x:T_xM\times T_xM\to \mathbb R\right).
\end{align}
A Lorentzian manifold differs from a Riemannian manifold by the signature of its bilinear form. A Riemannian metric has positive definite signature, that is, $g_x(Y,Y)>0$ for all $x\in M$ and nonzero $Y\in T_xM$. A Lorentzian metric has Lorentzian signature, that is, the form is still nondegenerate but the subspace of $T_xM$ on which $g_x$ is positive definite has dimension $N-1$, where $N$ is the dimension of the manifold.
An example of a Lorentzian metric is the Minkowski metric $\eta = -(\text{d}x^0)^2 + \sum_{i=1}^{n-1} (\text{d}x^i)^2$, given in coordinates $(x^0,x^1,\dots,x^{N-1})$.
A Finsler space is a generalization of a Riemannian manifold in the following way. One may define a Finsler space as a smooth manifold endowed with a smooth assignment of inner products to all of its tangent spaces, varying not only with position but also with direction:
\begin{align}
TM\setminus 0\ni X\mapsto \left(g_X:T_{\pi(X)}M\times T_{\pi(X)}M\to \mathbb R\right)
\end{align}
such that $g_{\lambda X} = g_X$ for all $\lambda>0$ (`homogeneity'). Here $\pi:TM\to M$ is the canonical projection. The homogeneity requirement is needed to ensure that the length of curves does not depend on the parameterization.
Any Riemannian manifold is also a Finsler manifold, but Finsler manifolds are more general. 
Usually a Finsler metric is defined not in terms of its metric tensor but in terms of a so-called Finsler function $F:TM\setminus 0\to [0,\infty)$. You can look up this definition in any standard textbook on Finsler geometry but it is equivalent to the one given above. The relation between $F$ and $g$ is given by
\begin{align}
F(X) = \sqrt{g(X,X)},\qquad g_X(Y,Z) = \frac{\text{d}}{\text{d} s}\frac{\text{d}}{\text{d} t}\bigg|_{t=s=0}\left[\frac{1}{2}F^2(X+s Y  + t Z)\right].
\end{align}
An example of such a Finsler function is the Randers metric (although dubbed Randers metric, it is not a metric tensor, it is a Finsler function)
\begin{align}
F(X) = \sqrt{\bar g (X,X)} + \omega(X),
\end{align}
where $\bar g$ is a Riemannian metric on $M$ and $\omega$ a 1-form on $M$ satisfying $||\omega||\equiv\sqrt{\bar g (\omega,\omega)}<1$. Smooth manifolds endowed with a Randers metric are called Randers spaces.
One more remark, I mentioned Minkowski space above as an example of a Lorentzian manifold. In Finsler geometry one also talks about Minkowski spaces but these are entirely different things. The Finsler function is defined such that on each tangent space, it restricts to a so-called Minkowski norm, which is a general notion of a norm. (Also here, a Minkowski norm is completely different from the Minkowski metric mentioned above, the terminology is a bit confusing, unfortunately.) Therefore it is sometimes said that, for a Finsler space, each $T_xM$ is a Minkowski space. This is not to be confused with the Lorentzian Minkowski space.
