Developed further in the study of riemannian geometry and various generalizations of the latter in the present book the tensor calculus of cuclidean 3 space is developed and then generalized so as to apply to a riemannian space of any number of dimensions. The purpose of this book is to bridge the gap between differential geometry of euclidean space of three dimensions and the more advanced work on differential geometry of generalised space the subject is treated with the aid of the tensor calculus which is associated with the names of ricci and levi civita and the book provides an introduction both to this calculus and to riemannian geometry. An introduction to riemannian geometry and the tensor calculus principal directions for a symmetric covariant tensor of the second order 47 euclidean space of n dimensions 50 examples iii 53 an introduction to riemannian geometry and the tensor calculus publisher. Introduction to riemannian geometry and the tensor calculus called the ricci principal directions of the space this case is however unique let t a be the components in the y s of the unit tangent t to a congruence of curves in v n in the following pages quadratic differential forms will play a prominent role