This paper investigates the trajectory tracking problem for quadrotor with attitude finite-time convergence via geometric approach. First, a global geometric dynamic description is presented on the special Euclidean group (SE(3)), and the trajectory tracking control is decomposed into two cascaded tracking control loops: the position tracking control loop and the attitude tracking control loop. Then, based on the fact that the attitude tracking loop is a fast loop, a finite-time controller based on the exponential coordinate is proposed to speed up the response rate of the attitude control loop, so that the artificial singularity and redundancy can be avoided. In addition, a backstepping controller is designed for the position tracking loop to construct the thrust magnitude control input for the position dynamics and the reference rotation matrix for the attitude tracking loop. Finally, the numerical simulation results are presented to demonstrate the effectiveness of this trajectory tracking strategy.