Semi-Riemannian Geometry

Chapter 5 Tensors on Vector Spaces

5.1 Tensors

Let V be a vector space, and let r, s ≥ 1 be integers. Following Section B.5 and Section B.6, we denote by Mult(V* ^r × V ^s , ℝ) the vector space of ℝ‐multilinear functions from V* ^r × V ^s to ℝ, where addition and scalar multiplication are defined as follows: for all functions 𝒜, ℬ, in Mult(V* ^r × V ^s , ℝ) and all real numbers c,

and

for all covectors η ^r , … , η ^r in V* and all vectors v ₁, … , v _s in V. For brevity, let us denote

We refer to a function 𝒜 in as an ( r, s )‐tensor, an r‐contravariant‐s‐covariant tensor, or simply a tensor on V, and we define the rank of to be (r, s). When s = 0, 𝒜 is said to be an r‐contravariant tensor or just a contravariant tensor, and when r = 0, 𝒜 is said to be an s‐covariant tensor or simply a covariant tensor. In Section 1.2, we used the term covector to describe what we now call a 1‐covariant tensor. Note that the sum of tensors is defined only when they have the same rank. The zero element of , called the zero tensor and denoted by 0, is the tensor that ...