ABSTRACT

This chapter defines the basic elements of differential geometry (vector, one-form, and tensor fields, coordinate transformations, metric spaces, etc.) in a more rigorous way. It denotes the vectors of the coordinate basis, associated with the coordinate system. The chapter finds the transformation rules for tensor components and tensor bases, under a coordinate transformation. The simplest way to transform the metric tensor under a coordinate transformation consists in deriving the transformation of the basis one-forms by differentiating the coordinate transformation, and then replacing them in the metric. Consequently the metric tensor also maps one-forms into vectors. The metric tensor is a real, linear function of two vectors, i.e. it takes two vectors and associates a real number to them, which is their scalar product.