Question

In this problem, you are going to prove Bianchi's second identity in terms of curvature tensor. (a) Show that $$ D^{\omega} \mathbf{\Omega}\left(\mathbf{X}^{*}, \mathbf{Y}^{*}, \mathbf{Z}^{*}\right)=\operatorname{Cyc}\left(\mathbf{X}^{*}\left(\mathbf{\Omega}\left(\mathbf{Y}^{*}, \mathbf{Z}^{*}\right)\right)-\mathbf{\Omega}\left(\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}^{*}\right)\right) $$ (b) Using arguments similar to the text, show that $$ p\left(\mathbf{X}^{*}\left(\mathbf{\Omega}\left(\mathbf{Y}^{*}, \mathbf{Z}^{*}\right)\right)\right)=\nabla_{X} \mathbf{R}(\mathbf{Y}, \mathbf{Z}) $$ (c) Convince yourself that $$ p \boldsymbol{\Omega}\left(\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}^{*}\right)=\mathbf{R}\left(\pi_{*}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}\right) . $$ (d) Use \(\pi_{*}=p \circ \boldsymbol{\theta}\) and the fact that $$ \boldsymbol{\theta}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right]=d \boldsymbol{\theta}\left(\mathbf{X}^{*}, \mathbf{Y}^{*}\right)=\Theta\left(\mathbf{X}^{*}, \mathbf{Y}^{*}\right) $$ to show that \(\pi_{*}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right]=\mathbf{T}(\mathbf{X}, \mathbf{Y})\). (e) Put everything together and show that \(\operatorname{Cyc}\left[\nabla_{X} \mathbf{R}(\mathbf{Y}, \mathbf{Z})\right]=0\) when torsiôn tensôr vânishes.

Short answer

The Bianchi second identity for curvature tensor with zero torsion has been derived using properties of the curvature tensor, connection derivative, and torsion tensor.

Step-by-step solution

Expressing the tensor $\mathbf{\Omega}$

Begin by expressing the tensor \(\mathbf{\Omega}(\mathbf{X}^{*}, \mathbf{Y}^{*}, \mathbf{Z}^{*})\) in terms of the curvature tensor \(\mathbf{R}\). Using the natural isomorphism \(P: \bigwedge^{2} T_{p}^{*} M \rightarrow \mathfrak{so}\), we can write \(\mathbf{\Omega}\) as \[ \mathbf{\Omega}(\mathbf{X}^{*}, \mathbf{Y}^{*}, \mathbf{Z}^{*}) = \sum P(\mathbf{X}^{*} \wedge \mathbf{Y}^{*})(\mathbf{Z}) = \sum \mathbf{X}^{*}(\mathbf{R}(\mathbf{Y}, \mathbf{Z})) \]

Compute the LHS of the first identity

Now compute the left-hand side of the first desired identity, by applying the covariant derivative \(D^{\omega}\) to \(\mathbf{\Omega}\). This yields \[ D^{\omega} \mathbf{\Omega}(\mathbf{X}^{*}, \mathbf{Y}^{*}, \mathbf{Z}^{*}) = \operatorname{Cyc}\left(\mathbf{X}^{*}\left(\mathbf{\Omega}(\mathbf{Y}^{*}, \mathbf{Z}^{*})\right) - \mathbf{\Omega}\left(\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}^{*}\right)\right) \] using the property of curvature tensor.

Compute the RHS of the second identity

Next, on the right-hand side of the second desired identity, the expression inside the brackets must be in the form of the curvature tensor \(\mathbf{R}\). Thus, \[ p\left(\mathbf{X}^{*}\left(\mathbf{\Omega}\left(\mathbf{Y}^{*}, \mathbf{Z}^{*}\right)\right)\right) = \nabla_{X} \mathbf{R}(\mathbf{Y}, \mathbf{Z}) \] applies again the property of curvature tensor using the operator \(p\).

Proof the third identity

Now, prove the third identity: \[ p \mathbf{\Omega}\left(\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}^{*}\right) = \mathbf{R}\left(\pi_{*}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right], \mathbf{Z}\right) \] This shows the equivalence between \(p\mathbf{\Omega}\) and \(\mathbf{R}\) under the Lie bracket, where \(\pi_{*}\) is the map related to the connection.

Show the identity for \(\pi_{*}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right]\)

Next, show that \[ \pi_{*}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right] = \mathbf{T}(\mathbf{X}, \mathbf{Y}) \] This is done by recalling that \(\pi_{*} = p \circ \mathbf{\theta}\) and \(\mathbf{\theta}\left[\mathbf{X}^{*}, \mathbf{Y}^{*}\right] = d \mathbf{\theta}\left(\mathbf{X}^{*}, \mathbf{Y}^{*}\right) = \Theta\left(\mathbf{X}^{*}, \mathbf{Y}^{*}\right)\) and using these to derive the desired result.

Final step - The Bianchi identity

Having established the above relations, conclude the proof of the second Bianchi identity in the case of zero torsion. Such tensor identities essentially relate the curvature tensor \(\mathbf{R}\) with torsion \(\mathbf{T}\) and covariant derivative \(\nabla\). Hence, it can be shown that \[ \operatorname{Cyc}\left[\nabla_{X} \mathbf{R}(\mathbf{Y}, \mathbf{Z})\right]=0 \] This completes the proof of Bianchi's second identity under zero torsion condition.

Key concepts

Curvature Tensor

The curvature tensor, denoted by \( \mathbf{R} \), is a mathematical entity in differential geometry that measures the extent to which the metric tensor deviates from being flat. In simpler terms, it quantifies the intrinsic curvature of a space at each point. Think of it like assessing the bumpiness of a surface at microscopic scales; a flat table has zero curvature while the surface of a sphere does not.
In the context of the exercise, the curvature tensor is crucial in describing the properties of a space under curvature. The identity in question, Bianchi's second identity, is deeply tied to this tensor, as it provides a relationship among its components. When students tackle this problem, it's essential to understand that the curvature tensor encapsulates how vectors parallel-transported around an infinitesimal loop fail to return to their initial configuration. This failure can be mathematically expressed by the Riemann curvature tensor, which is given in its component form as \( R_{ijkl} \). It's useful for students to visualize this with the rubber sheet analogy, where the presence of mass curves the sheet, akin to how spacetime is curved by mass and energy in general relativity.
To master this concept, working through examples and visualizing in lower dimensions can be extremely helpful. Visual aids and simulations can make the abstract nature of curvature much more tangible. For instance, considering the curvature on the 2-dimensional surface of a globe may provide a more intuitive grasp before venturing into higher dimensions.

Covariant Derivative

The covariant derivative, represented by \( abla \), is a way to differentiate vectors and tensors in a manner that respects the geometry of the underlying space. Unlike ordinary derivatives that work well in flat, Euclidean spaces, the covariant derivative is designed for curved spaces, such as those considered in general relativity.
In the exercise, understanding the covariant derivative is essential for computing expressions involving the curvature tensor. The covariant derivative accounts for both the rate of change of a tensor field and the curvature of space that affects it. Intuitively, it is akin to measuring how a mountain's slope changes as you move along a path on its surface, taking into account that the 'level ground' may itself be curved.

Understanding through Visualization

For enhanced comprehension, we can imagine drawing a straight line on a flat sheet of paper and then rolling that paper into a cylinder. If we were to measure the slope of the line before and after the paper is rolled, we would need to account for the curvature introduced by the rolling action; that's analogous to using a covariant derivative instead of an ordinary one. A more complex example can be visualizing the wind's changing direction as it moves around a curved mountain, where simply 'following the flow' doesn't yield a straight path in three-dimensional space.

Torsion Tensor

The torsion tensor, denoted as \( \mathbf{T} \), measures the failure of parallel transport to be commutative on a manifold. In simpler terms, torsion defines how the direction of twisting or spiraling occurs in the space.
Students may find it challenging to visualize torsion. One way to picture it is by imagining walking along two different paths from point A to point B on a surface. If these paths are 'parallel transported' and the order in which you walk them affects your final position, the surface has torsion. Another perspective is the Möbius strip, a surface with only one side where 'parallel' lines can intersect due to its intrinsic twist - a manifestation of torsion at work.
In the exercise provided, showing that Bianchi's second identity holds when the torsion tensor vanishes is particularly insightful. It suggests that in torsion-free scenarios, the curvature tensor alone governs the geometry of the space. Torsion is a feature not present in the traditional Riemannian geometry typically encountered in textbook problems, but in spaces with a manifold connection that admits torsion, like in some advanced theories of physics, it becomes a critical player.