"This text follows a course given in the first semester 1925-1926 in the Faculty of Sciences of the University of Paris. I have adopted, in principle, the same point of view as in the text on the same subject that is Included in the series Memorial dee Sciences Mathemtztiquea (No. IX). I have almost always assumed given a system of local coordinates and a metric tensor for the space in question: This has required Ideas from tensor analysis. I have tried to do this while emphasizing the essential qeonetrlc features and keeping in close contact with Euclidean geometry. The utility of the absolute differential calculus of Ricci and Levi-Civita must be tempered by an avoidance of exclusively formal calculations, where the debauch of Indices disguises an often very simple geometric reality. It is this reality that I have sought to reveal." - from the preface to the first edition.