Metric Tensor

the set of quantities that define the geometric properties of a space (the metric of the space). In the general case of an n -dimensional Riemannian space, the metric is defined as the square of the distance ds²Pointe (x¹,x²,.......,xⁿ) and (x¹ + dx¹, x² + dx²,...., xⁿ)+ dxⁿ)

where x^l, x², . . . , xⁿ are coordinates and the gik are certain functions of the coordinates. The set of the quantities gik forms a second-rank tensor, which is called the metric tensor. This tensor is symmetric, that is, gik = gki The form of the components of the metric tensor gik depends on the choice of the coordinate system, but ds² does not change in changing from one coordinate system to another, that is, it is invariant with respect to transformations of coordinates. If the metric tensor can be reduced to the form

between two infinitesimally close by selection of the coordinate system, then the space is a plane, Euclidean space. (For a three-dimensional space, ds² = dx² + dy² + dz², where x¹ = x, x² = y, and x³ = z are the rectangular Cartesian coordinates.) If a metric tensor cannot be reduced to the form (2) by any transformation of coordinates, then the space is curved and the curvature of the space is defined by the metric tensor. In the theory of relativity, the space-time metric is defined by a metric tensor.