In this work, analysis of the space-time manifold, their kinematic groups and Lie algebras are made intuitive as far as possible. First of all, from the analysis of the iner-tial frames it is shown that according to the Beltrami theorem in Riemann Geometry, the space-time manifold, in which there exists global inertial frame, should be a pseudo-sphere. So that the kinematic group must be a rotation group, thus the explicity analy-tical expressions of such kinematical transformations and the commutative relations among the corresponding generators can be formulated easily. Consequently, the con-tractions of such manifolds, kinematic groups and Lie algebras can be deduced concretely and intuitively.