The Finsler space is endowed with the Matsumoto connection \(M\Gamma =(\Gamma^ i_{jk},\Gamma^ i_ k,C^ i_{jk})\) where \(\Gamma^ i_ k=G^ i_ k+T^ i_ k\), \(\Gamma^ i_{jk}=\Gamma^ j_{k\| j}+Q^ i_{jk}\), \(T^ i_ k\) and \(Q^ i_{jk}\) are tensors. Several kinds of connection coefficients which are obtained for different T and Q (for example \(T^ i_ 0=T^ 0_ k=0\), \(Q_{i0k}+D_{jk}=0\), \(T^ i_ k=f(x,y)L^ 3(x,y)C^ iC_ k\), \(T^ i_ k=0\), \(\Gamma^ i_{jk}=\Gamma^{*i}_{jk}+f(l_ j\delta^ i_ k-l^ ig_{jk}))\) and for different conditions concerning the h and v metrical properties and different kinds of torsion tensors are given.
It is examined when M with respect to these different kinds of connections is of scalar curvature, constant curvature, h flat, (h,hv) flat, v-flat, n-flat, quasi Minkowskian, locally Minkowskian or locally Euclidean. The torsion and curvature tensors of M for these connections are also given.