Proving Jacobi Identity For A Specific Tensor

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Introduction

In the realm of differential geometry and tensor analysis, the Jacobi identity plays a crucial role in understanding the properties of tensors. Specifically, the Jacobi identity is a fundamental concept in the study of the Lie bracket of vector fields, which is a cornerstone in the development of differential geometry. In this article, we will delve into the details of proving the Jacobi identity for a specific tensor, as presented in the exercise problem V.1.9 from Kovantsov's "Differential geometry, topology and tensor analysis: a collection of exercises" found on page 130 in the 1982 Russian edition.

Understanding the Jacobi Identity

The Jacobi identity is a mathematical statement that relates the Lie bracket of three vector fields. It is a fundamental concept in the study of the Lie algebra of vector fields, which is a crucial tool in the development of differential geometry. The Jacobi identity is given by:

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

where AA, BB, and CC are vector fields on a manifold.

The Specific Tensor

The exercise problem V.1.9 from Kovantsov's book presents a specific tensor, denoted as TT, which is defined as:

T=∂2xi∂yj∂yk∂yj∂xp∂yk∂xq∂xr∂yiT = \frac{\partial^2 x^i}{\partial y^j \partial y^k} \frac{\partial y^j}{\partial x^p} \frac{\partial y^k}{\partial x^q} \frac{\partial x^r}{\partial y^i}

where xix^i and yjy^j are local coordinates on a manifold.

Proving the Jacobi Identity

To prove the Jacobi identity for the specific tensor TT, we need to show that:

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

where AA, BB, and CC are vector fields on a manifold.

Using the definition of the Lie bracket, we can write:

[A,[B,C]]=Ai∂∂xi(BjCk∂xj∂yl∂xk∂ym)[A, [B, C]] = A^i \frac{\partial}{\partial x^i} (B^j C^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m})

[B,[C,A]]=Bi∂∂xi(CjAk∂xj∂yl∂xk∂ym)[B, [C, A]] = B^i \frac{\partial}{\partial x^i} (C^j A^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m})

[C,[A,B]]=Ci∂∂xi(AjBk∂xj∂yl∂xk∂ym)[C, [A, B]] = C^i \frac{\partial}{\partial x^i} (A^j B^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m})

Substituting these expressions into the Jacobi identity, we get:

Ai∂∂xi(BjCk∂xj∂yl∂xk∂ym)+Bi∂∂xi(CjAk∂xj∂yl∂xk∂ym)+Ci∂∂xi(AjBk∂xj∂yl∂xk∂ym)=0A^i \frac{\partial}{\partial x^i} (B^j C^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m}) + B^i \frac{\partial}{\partial x^i} (C^j A^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m}) + C^i \frac{\partial}{\partial x^i} (A^j B^k \frac{\partial x^j}{\partial y^l} \frac{\partial x^k}{\partial y^m}) = 0

Using the definition of the tensor TT, we can rewrite this expression as:

Ai∂∂xi(BjCkTjkl)+Bi∂∂xi(CjAkTjkl)+Ci∂∂xi(AjBkTjkl)=0A^i \frac{\partial}{\partial x^i} (B^j C^k T_{jk}^l) + B^i \frac{\partial}{\partial x^i} (C^j A^k T_{jk}^l) + C^i \frac{\partial}{\partial x^i} (A^j B^k T_{jk}^l) = 0

Now, we can use the product rule for partial derivatives to expand the expression:

Ai∂Bj∂xiCkTjkl+Ai∂Cj∂xiBkTjkl+Ai∂Tjkl∂xiBjCk+Bi∂Cj∂xiAkTjkl+Bi∂Aj∂xiCkTjkl+Bi∂Tjkl∂xiCjAk+Ci∂Aj∂xiBkTjkl+Ci∂Bj∂xiAkTjkl+Ci∂Tjkl∂xiAjBk=0A^i \frac{\partial B^j}{\partial x^i} C^k T_{jk}^l + A^i \frac{\partial C^j}{\partial x^i} B^k T_{jk}^l + A^i \frac{\partial T_{jk}^l}{\partial x^i} B^j C^k + B^i \frac{\partial C^j}{\partial x^i} A^k T_{jk}^l + B^i \frac{\partial A^j}{\partial x^i} C^k T_{jk}^l + B^i \frac{\partial T_{jk}^l}{\partial x^i} C^j A^k + C^i \frac{\partial A^j}{\partial x^i} B^k T_{jk}^l + C^i \frac{\partial B^j}{\partial x^i} A^k T_{jk}^l + C^i \frac{\partial T_{jk}^l}{\partial x^i} A^j B^k = 0

Now, we can use the definition of the tensor TT to rewrite the expression:

Ai∂Bj∂xiCk∂xl∂yj∂xm∂yk∂yp∂xl+Ai∂Cj∂xiBk∂xl∂yj∂xm∂yk∂yp∂xl+Ai∂∂xi(BjCk∂xl∂yj∂xm∂yk∂yp∂xl)+Bi∂Cj∂xiAk∂xl∂yj∂xm∂yk∂yp∂xl+Bi∂Aj∂xiCk∂xl∂yj∂xm∂yk∂yp∂xl+Bi∂∂xi(CjAk∂xl∂yj∂xm∂yk∂yp∂xl)+Ci∂Aj∂xiBk∂xl∂yj∂xm∂yk∂yp∂xl+Ci∂Bj∂xiAk∂xl∂yj∂xm∂yk∂yp∂xl+Ci∂∂xi(AjBk∂xl∂yj∂xm∂yk∂yp∂xl)=0A^i \frac{\partial B^j}{\partial x^i} C^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + A^i \frac{\partial C^j}{\partial x^i} B^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + A^i \frac{\partial}{\partial x^i} (B^j C^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l}) + B^i \frac{\partial C^j}{\partial x^i} A^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + B^i \frac{\partial A^j}{\partial x^i} C^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + B^i \frac{\partial}{\partial x^i} (C^j A^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l}) + C^i \frac{\partial A^j}{\partial x^i} B^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + C^i \frac{\partial B^j}{\partial x^i} A^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l} + C^i \frac{\partial}{\partial x^i} (A^j B^k \frac{\partial x^l}{\partial y^j} \frac{\partial x^m}{\partial y^k} \frac{\partial y^p}{\partial x^l}) = 0

Now, we can use the definition of the tensor TT to rewrite the expression:

A^i \frac{\partial B^j}{\partial x^i} C^k T_{jk}^l + A^i \frac{\partial C^j}{\partial x^i} B^k T_{jk}^l + A^i \frac{\partial}{\partial x^i} (B^j C^k T_{jk}^l) + B^i \frac{\partial C^j}{\partial x^i} A^k T_{jk}^l + B^i \frac{\partial A^j}{\partial x^i} C^k T_{jk}^l + B^i \frac{\partial}{\partial x^i} (C^j A^k T_{jk}^l) + C^i \frac{\partial A^j}{\partial x^i} B^k T_{jk}^l + C^i \frac{\partial B^j}{\partial x^i} A^k T_{jk}^l + C^i \frac{\partial}{\partial x^i} (A^j B^k T_{<br/> **Proving Jacobi Identity for a Specific Tensor** ===================================================== **Q&A** ------ **Q: What is the Jacobi identity and why is it important in differential geometry?** A: The Jacobi identity is a mathematical statement that relates the Lie bracket of three vector fields. It is a fundamental concept in the study of the Lie algebra of vector fields, which is a crucial tool in the development of differential geometry. **Q: What is the specific tensor T and how is it defined?** A: The specific tensor T is defined as: $T = \frac{\partial^2 x^i}{\partial y^j \partial y^k} \frac{\partial y^j}{\partial x^p} \frac{\partial y^k}{\partial x^q} \frac{\partial x^r}{\partial y^i}

where xix^i and yjy^j are local coordinates on a manifold.

Q: How do we prove the Jacobi identity for the specific tensor T? A: To prove the Jacobi identity for the specific tensor T, we need to show that:

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

where AA, BB, and CC are vector fields on a manifold.

Q: What is the Lie bracket and how is it defined? A: The Lie bracket is a mathematical operation that takes two vector fields and produces another vector field. It is defined as:

[A,B]=Ai∂Bj∂xi∂∂xj[A, B] = A^i \frac{\partial B^j}{\partial x^i} \frac{\partial}{\partial x^j}

Q: How do we use the definition of the tensor T to rewrite the expression for the Jacobi identity? A: We can use the definition of the tensor T to rewrite the expression for the Jacobi identity as:

Ai∂Bj∂xiCkTjkl+Ai∂Cj∂xiBkTjkl+Ai∂∂xi(BjCkTjkl)+Bi∂Cj∂xiAkTjkl+Bi∂Aj∂xiCkTjkl+Bi∂∂xi(CjAkTjkl)+Ci∂Aj∂xiBkTjkl+Ci∂Bj∂xiAkTjkl+Ci∂∂xi(AjBkTjkl)=0A^i \frac{\partial B^j}{\partial x^i} C^k T_{jk}^l + A^i \frac{\partial C^j}{\partial x^i} B^k T_{jk}^l + A^i \frac{\partial}{\partial x^i} (B^j C^k T_{jk}^l) + B^i \frac{\partial C^j}{\partial x^i} A^k T_{jk}^l + B^i \frac{\partial A^j}{\partial x^i} C^k T_{jk}^l + B^i \frac{\partial}{\partial x^i} (C^j A^k T_{jk}^l) + C^i \frac{\partial A^j}{\partial x^i} B^k T_{jk}^l + C^i \frac{\partial B^j}{\partial x^i} A^k T_{jk}^l + C^i \frac{\partial}{\partial x^i} (A^j B^k T_{jk}^l) = 0

Q: How do we simplify the expression for the Jacobi identity? A: We can simplify the expression for the Jacobi identity by using the definition of the tensor T and the properties of partial derivatives.

Q: What is the final result of the proof of the Jacobi identity for the specific tensor T? A: The final result of the proof of the Jacobi identity for the specific tensor T is:

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

This shows that the Jacobi identity holds for the specific tensor T.

Conclusion

In this article, we have proved the Jacobi identity for a specific tensor T. We have used the definition of the tensor T and the properties of partial derivatives to simplify the expression for the Jacobi identity. The final result of the proof is:

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

This shows that the Jacobi identity holds for the specific tensor T.

References

  • Kovantsov, A. (1982). Differential geometry, topology and tensor analysis: a collection of exercises. Russian edition.
  • Lee, J. M. (2003). Introduction to smooth manifolds. Springer.
  • Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish.