^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.24 - Components of Tensors and the Outer Product|2.24 Components of Tensors and the Outer Product]] | [[Schutz - 2.26 - Basis Transformations|2.26 Basis Transformations]] >> # Contraction *Contraction* is the operation of summing over repeated indices. For example, given the basis of a $(2,0)$ tensor shown in [[Schutz - 2.24 - Components of Tensors and the Outer Product#^9608f9|§2.25]], $\{V^i \omega_j\}$ By summing on the index, we get a $(0,0)$ tensor, i.e. a scalar, $V^j \omega_j$. Similar contractions apply by summing over repeated indices of higher-order tensors. Contractions are basis independent as prove on page 60 of the book.