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Chapter 15: Problem 24

Show that the Lie derivative \(\mathcal{L}_{x}\) commules with all operations of contraction \(C_{t}^{y}\) on a tensor field \(T\), $$ \mathcal{L}_{X} C_{J} T=C_{j} \mathcal{L}_{\mathrm{Y}} T $$

Short Answer

Expert verified
Therefore, it's proven that the Lie derivative \(\mathcal{L}\) commutes with all operations of contraction \(C_{J}\) on a tensor field \(T\).

Step by step solution

01

Definition of the Lie derivative

The Lie derivative \(\mathcal{L}_X\) of a tensor field T in direction X is given by the limit: \[ \mathcal{L}_XT = \lim_{t→0} \frac{\phi_t^*T - T}{t} \] Where \(\phi_t\) is the flow of X
02

Apply contraction to both sides of the Lie derivative definition

Now apply contraction \(C_J\) to both sides of this definition to get: \[ C_J(\mathcal{L}_XT) = \lim_{t→0} \frac{C_J(\phi_t^*T - T)}{t} \]
03

Applying the Leibnitz rule

Now, we can apply the Leibnitz rule (which states that contraction operations distribute over addition and scale with scalar multiplication) on the right hand side of this equation: \[ C_J(\mathcal{L}_XT) = \lim_{t→0} \frac{C_J(\phi_t^*T)-C_J(T)}{t} \]
04

Recognizing the Lie derivative

Now one can observe that the right hand side is just the definition of the Lie derivative applied to contraction of the tensor field \(T\), hence: \[ C_J(\mathcal{L}_XT) = \mathcal{L}_X(C_JT) \]

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Most popular questions from this chapter

Show that the real projective \(n\)-space \(P^{\text {* }}\) defined in Example \(10.15\) as the set of straight lines through the origin in \(\mathbb{R}^{n+1}\) is a differentiable manifold of dimension \(n\), by finding an atlas of compatible charts that cover it.

Show that the set of real \(m \times m\) matrices \(M(m, n ; \mathbb{R})\) is a manifold of dimension \(m n\). Show that the matrix multiplication map \(M(m, k ; \mathbb{R}) \times M(k, n ; \mathbb{R}) \rightarrow M f(m, n ; \mathbb{R})\) is differentiable.

On the \(n\)-sphere \(S^{n}\) find coordinates corresponding to (i) stereographic projection, (ii) spherical polars.

Show that the map \(\alpha: \hat{R}^{2} \rightarrow \mathbb{R}^{3}\) defined by $$ u=x^{2}+y^{2}, \quad v=2 x y, \quad w=x^{2}-y^{2} $$ is an immersion Is it an embedded submanifold?

For any real positive number \(n\) show that the vector field \(X=x^{n} \partial_{x}\) is differentiable on the manifold \(\mathbb{R}^{+}\)consisting of the positive real line \([x \in \mathbb{R} \mid x>0]\). Why is this not true in general on the enture real line \(\mathrm{R}\) ? As done for the case \(n=2\) in Example 15,13, find the maximal one-parameter subgroup \(\sigma_{i}\) generated by this vector field at any point \(x>0\).
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