A main core of three topics: Li Yau type differential Harnack inequalities, entropy formulas and space-time geodesics, is presented in this book. Emphasis is put on the connections between these three subjects. Each topic is first explained for the heat equation on a static manifold. The results are then compared with the results for the Ricci flow equation or a heat equation on a manifold evolving by Ricci flow. The presented theory is recent, mainly connected with R. S. Hamilton, P. Li, L. Ni, G. Perelman and S. T. Yau. Applications for three-dimensional manifolds towards the Poincaré conjecture are not considered. The book is comprehensible and well-ordered. The main ideas of proofs are well explained; however, some parts of the book are rather technical. As the author presents the core of the theory in detail, the book is suitable for advanced students or non-experts familiar with the basic concepts and notions of Riemannian geometry. No preliminary knowledge of Ricci flow or Harnack inequalities is required.