The Lie Derivative
We first introduce some tools and facts on integral curves and flows of vector fields. Then mappings of tensor fields, in particular by diffeomorphisms, are defined. This allows us to introduce the Lie derivative of a tensor field with respect to a vector field. The properties of this important differential operation are formulated in several theorems, most of which will be proven later in Chap. 15. We also provide coordinate expressions of the Lie derivative. Killing fields of a (pseudo-) Riemannian manifold play an important role in GR, because they induce isometries.
KeywordsVector Field Riemannian Manifold Uniqueness Theorem Product Rule Linear Extension
Mathematical Tools: Selection of Mathematical Books
- 47.S. Lang, Differential and Riemannian Manifolds. Graduate Texts in Mathematics, vol. 160 (Springer, New York, 1995) Google Scholar