TITLE: General Relativity and Differential Geometry intuitions of Second Bianchi Identity
QUESTION [13 upvotes]: In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-


$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$


It is said that this identity amounts, in the Riemmanian geometry sense, to the saying that, loosely, "the boundary of the bounday is null". So my questions are
Question 1: What would be a more rigorous statement of "the boundary of the boundary is null" and how does the above identity is equivalent ot it?
Question 2: Is there an analog to this geometrical intuition in general relativity?
I assume both questions might be answered with a reference, which is even better.

REPLY [5 votes]: The second Bianchi identity can also be viewed as a consequence of the diffeomorphism invariance of the Riemann curvature tensor, i.e. the fact that $\operatorname{Rm}_{\varphi^*g} = \varphi^* \operatorname{Rm}_g$ for any metric $g$ and diffeomorphism $\varphi$. Differentiating with respect to $t$ the equation $\operatorname{Rm}_{\varphi_t^*g} = \varphi_t^* \operatorname{Rm}_g$ , where $\{\varphi_t\}$ is a one-parameter family of diffeomorphisms, yields a proof of the first and second Bianchi identities.
This is discussed in (for instance) Chapter 5.3.1 of The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem by Ben Andrews and Christopher Hopper.
In terms of a relativistic interpretation, this is related to the idea of general covariance, which requires the laws of physics to be diffeomorphism invariant.