The invariant geometric structures on the statistical manifold under sufficient statistics have played an important role in both statistical inference and information theory. In this paper, we focus on one of the commonly used invariant geometric structures, the Amari-Chentsov structure, on a statistical manifold. The manifold is derived from statistical models for accelerated life tests (ALTs) with censoring based on the exponential family of distributions. The constant-stress ALTs and step-stress ALTs are considered. We show that the statistical manifold still belongs to the exponential family of distributions, but the cumulant generating function depends on a random variable related to the experimental design of the ALT, which is different from the usual situation. We also investigate the Bregman divergence and Riemannian metric. The relationships between the Riemannian metric and the expected Fisher information metric are studied. The dual coordinate system is studied by using the Legendre transformation. Then, the Amari-Chentsov structure is derived based on the two different coordinate systems. The methodologies are illustrated by using two distributions, the exponential and gamma distributions, in the exponential family of distributions. Finally, using the results of Fisher information metric, optimal designs of the two types of ALTs are presented with different optimal criteria. Finally, numerical examples are provided to demonstrate the practical applications of the results developed in this paper..