The Lie Derivative

  • Norbert Straumann
Part of the Graduate Texts in Physics book series (GTP)


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.


Vector Field Riemannian Manifold Uniqueness Theorem Product Rule Linear Extension 
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Mathematical Tools: Selection of Mathematical Books

  1. 47.
    S. Lang, Differential and Riemannian Manifolds. Graduate Texts in Mathematics, vol. 160 (Springer, New York, 1995) Google Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Norbert Straumann
    • 1
  1. 1.Mathematisch-Naturwiss. Fakultät, Institut für Theoretische PhysikUniversität ZürichZürichSwitzerland

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