Computer geometry and encoding the information on a manifold

In this review, we try to answer the following question why should one study differential geometry? First of all, differential geometry is a Jewel of Mathematics. It is a prerequisite for theoretical physics. Secondly, in recent years, new and important applications have been discovered. Surprisingly, the structures of differential geometry are ideally suited for coding theory, information geometry and imaging process, kinematics of Robotics and computer aided geometric design, optimization and so on The main goal of the review is to establish a bridge between the theoretical aspects of modern geometry and topology on the one hand and computer experimental geometry on the other. The flood of information through various computer networks such as the internet characterizes the world situation in which we live. Information words, often called virtual spaces and cyberspace, have been formed on computer networks. The complexity of information worlds has been increasing almost exponentially through the exponential growth of computer networks. Such nonlinearity in growth and in scope characterizes information words. In other words the characterization of nonlinearity is the key to understanding, utilizing and living with the flood of information. The characterization approach is by characteristic points such as peaks, pits, and passes, according to the Morse theory on the manifold. Another approach is by singularity signs such as folds, cusps bifurcation, nodes, butterfly and swallowtail. Atoms and molecules are the other fundamental characterization approach. Topology and geometry including differential topology, serve as the framework for the characterization.