The Riemannian Metric Two Tensor

We consider a covariant basis of vectors and then consider all possible dot products of these vectors. This will produce a symmetric two covariant tensors called the Riemannian metric tensor. We prove that this tensor does indeed satisfy the transformational law of a two covariant tensor, and then describe how it can be used to construct the raising and lowering operators. This also will allow us to construct a dual basis by raising the indices of a covariant basis. #mikethemathematician, #mikedabkowski, #profdabkowski, #tensoranalysis