## 2

## Tensor Analysis and Riemannian Geometry

#### Part I Line Element

##### 2.1 RIEMANNIAN SPACE

In the Euclidean space of three dimensions, each point is specified by three coordinates (*x*^{1}*, x*^{2}*, x*^{3}). The distance* ds* between two neighbouring points (*x*^{1}*, x*^{2}*, x*^{3}) and (*x*^{1} +* dx*^{1}*, x*^{2} +* dx*^{2}*,* *x*^{3} +* dx*^{3}) is given by

*ds*^{2} = (*dx*^{1})^{2} + (*dx*^{2})^{2} + (*dx*^{3})^{2 }(2.1)

We may extend the concept of Cartesian space in three dimensions to *n*-dimensional space. Each point will be designated by* n* coordinates (*x*^{1}*,x*^{2}*,…,x*^{n}), which are shown collectively by (*x*). Further, we assume that the distance between any two neighbouring points is given by

where* g*_{μν} (*x*) are functions of the coordinates ...