Jacobi identity for covariant derivatives proof.

In summary, the Jacobi identity for covariant derivatives is a mathematical relationship that describes the transformation of covariant derivatives under a change of coordinates. It is derived by manipulating the covariant derivatives and using the properties of vector fields and coordinate transformations. This identity is significant in differential geometry as it ensures that the covariant derivatives form a Lie algebra and allows for the development of important geometric concepts. It can also be extended to higher-order derivatives, known as the Bianchi identity, and is used in various practical applications such as studying curvature and torsion, and analyzing particles in curved space-time.
  • #1
center o bass
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Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that

$$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$

without going into coordinate basis.
 
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  • #2
Never mind. It's simply due to the properties of the commutator. The jacobi identity for lie brackets does not depend on on it being partial derivative operators; It can be any kind of operators.
 

1. What is the Jacobi identity for covariant derivatives?

The Jacobi identity for covariant derivatives is a mathematical relationship that describes how the covariant derivatives of a quantity transform under a change of coordinates. It is often used in differential geometry and is an important tool in studying manifolds and their properties.

2. How is the Jacobi identity for covariant derivatives derived?

The Jacobi identity for covariant derivatives is derived by applying the definition of the covariant derivative and using the properties of vector fields and coordinate transformations. The proof involves manipulating the covariant derivatives and using the commutator bracket to show that the identity holds true.

3. What is the significance of the Jacobi identity for covariant derivatives?

The Jacobi identity for covariant derivatives is significant because it ensures that the covariant derivatives of a vector field form a Lie algebra, which is a fundamental concept in differential geometry. It also allows for the development of important geometric concepts such as curvature and torsion.

4. Can the Jacobi identity for covariant derivatives be extended to higher-order derivatives?

Yes, the Jacobi identity for covariant derivatives can be extended to higher-order derivatives by using the definition of the covariant derivative for tensors. This extended identity is known as the Bianchi identity and is essential in studying the geometry of manifolds.

5. How is the Jacobi identity for covariant derivatives used in practical applications?

The Jacobi identity for covariant derivatives is used in various fields of science, such as physics, engineering, and mathematics. It is used to study the curvature and torsion of manifolds, to develop geometric concepts in general relativity, and to analyze the behavior of particles in curved space-time.

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