Semi-Riemannian Geometry

Chapter 6 Tensors on Scalar Product Spaces

6.1 Contraction of Tensors

In Section 5.4, we discussed contraction of tensors over vector spaces. In the setting of scalar product spaces, there is more to say.

Theorem 6.1.1 (Orthonormal Basis Expression for Ordinary Contraction).

Let (V, g) be a scalar product space with signature (&ip.eop; ₁ ,…,&ip.eop; _m ), let (e ₁ , …, e _m ) be an orthonormal basis for V, and let A be a tensor in . Then:

1. For integers s ≥ 2 and 1 ≤ l ≤ s – 1, the tensor in is given by

   

   for all vectors v ₁, …, v _s − 1 in V, where ℜ _s is given by (5.3.1).

2. For s = 1, the real number is given by

   

where ℜ is given by (5.3.2).

Proof

(a): By Theorem 5.4.5, the components of with respect to (e ₁, …, e _m ) are

(6.1.1) ...