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Chapter 25: Lie Derivative, Invariance and Killing Vector Fields

https://doi.org/10.1142/9789813275386_0025Cited by:0 (Source: Crossref)

Abstract:

Let M be an n-dimensional C^∞ differentiable manifold with local coordinates x_j, j = 1, …, n. Real valued C^∞ vector fields and real valued C^∞ differential forms on M can be considered. The components of the vector field V are denoted by V_j∂/∂x_j . This means

V : = V_{1} (x) \frac{\partial}{\partial x_{1}} + V_{2} (x) \frac{\partial}{\partial x_{2}} + \dots + V_{n} (x) \frac{\partial}{\partial x_{n}}

$V : = V_{1} (x) \frac{\partial}{\partial x_{1}} + V_{2} (x) \frac{\partial}{\partial x_{2}} + \dots + V_{n} (x) \frac{\partial}{\partial x_{n}}$

in local coordinates. The Lie derivative of a differential form α with respect to V is defined by the derivative of α along the integral curve t ↦ Φ_t of V , i.e.

L_{V} α : = \lim_{t \to 0} \frac{Φ_{t}^{*} α - α}{t} .

$L_{V} α : = \lim_{t \to 0} \frac{Φ_{t}^{*} α - α}{t} .$

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Chapter 25: Lie Derivative, Invariance and Killing Vector Fields

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