Riemannian geometry: a metric entrance (Q2718654)

As the title suggests, this book is an introduction to Riemannian geometry using a metric approach. Using a very elegant style and clarity the author succeeds in presenting the most important results of Riemannian geometry in less than 60 pages. The main concept used by the author is the Riemannian patch, which is nothing but an open subset of \(\mathbb R^n\) with a Riemannian metric. The geodesics defined as locally shortest curves are easily seen to exist. Then, the author introduces the notions of connection and exponential map starting from the theory of the first variation of arc-length formula (sections 3.5). Other local concepts studied are: isometries, Jacobi vector fields, Riemannian sectional curvature, etc. A key result which allows the author to make the passage from local to global Riemannian manifolds is that the isometries between Riemannian patches are smooth (see section 6).NEWLINENEWLINE One of the fundamental centerpieces in global Riemannian geometry is the Hopf-Rinov theorem; it is proved in section 12. This is followed by two of the most classical results concerning relations between the curvature and topology of Riemannian manifolds: the Hadamard-Cartan and the Bonnet-Myers theorems. The last sections concern: connections and differential forms, submanifold theory and relative curvature (second fundamental forms), models for space forms and Riemannian submersions (Gray-O'Neill formulas). Written by one of the leading experts in the field, this book can be a great reading for both beginners and advanced students. We highly recommend this book to all readers interested in Riemannian geometry.