Before we start I have again found a very important paragraph, it is the last in section 3.2 and describes how we have got from the beginning to here. The beginning was the concept of a set which became a manifold and we ended up with metrics and covariant derivatives!
After nearly getting to the end of chapter 3 I realised that my ideas about covariant derivatives needed refinement and that I did not really understand parallel transport. With the former it would seem that metric compatibility, ##\nabla_\mu g_{\lambda\nu}=0##, arises out of the Leibnitz rule and the demand that the covariant derivative is a tensor and is not an 'additional property' - with the caveat that the manifold in question has a metric with the usual properties. On the latter I was still hazy until I explored further in this review.
I still don't think I have mastered every detail of this chapter to section 8 on Killing vectors but one thing is clear: If you have a tensor field ##T\left(x^\mu\right)## which you can express in terms of coordinates ##x^\mu## and we consider two points ##x^\alpha,x^\beta## then if you parallel transport the tensor from ##x^\alpha## along a geodesic to ##x^\beta## and call it ##T\prime\left(x^\beta\right)## there, then in all likelihood ##T\prime\left(x^\beta\right)\neq T\left(x^\beta\right)## and in some sense at least ##T^\prime\left(x^\beta\right)=T\left(x^\alpha\right)##. I think. Equation (20) in the document is saying that for a vector.
The rest of chapter 3 was straightforward and finally we met the geodesic deviation equation $$
A^\mu=\frac{D^2}{dt^2}S^\mu=R_{\ \ \nu\sigma\rho}^\mu T^\nu T^\rho S^\sigma
$$##A^\mu## is the "relative acceleration of (neighbouring) geodesics", ##S^\mu## is a vector orthogonal to a geodesic (pointing towards its neighbour) and ##T^\mu## is a vector tangent to the geodesic. The equation expresses the idea that the acceleration between two neighbouring geodesics is proportional to the curvature and, physically, it is the manifestation of gravitational tidal forces.
The six page document repeats the above and reviews covariant derivatives, parallel transport and geodesics as shown in the video, Riemann and Killing. It's at Commentary 3 Review chapter 3.pdf.
I still don't think I have mastered every detail of this chapter to section 8 on Killing vectors but one thing is clear: If you have a tensor field ##T\left(x^\mu\right)## which you can express in terms of coordinates ##x^\mu## and we consider two points ##x^\alpha,x^\beta## then if you parallel transport the tensor from ##x^\alpha## along a geodesic to ##x^\beta## and call it ##T\prime\left(x^\beta\right)## there, then in all likelihood ##T\prime\left(x^\beta\right)\neq T\left(x^\beta\right)## and in some sense at least ##T^\prime\left(x^\beta\right)=T\left(x^\alpha\right)##. I think. Equation (20) in the document is saying that for a vector.
The rest of chapter 3 was straightforward and finally we met the geodesic deviation equation $$
A^\mu=\frac{D^2}{dt^2}S^\mu=R_{\ \ \nu\sigma\rho}^\mu T^\nu T^\rho S^\sigma
$$##A^\mu## is the "relative acceleration of (neighbouring) geodesics", ##S^\mu## is a vector orthogonal to a geodesic (pointing towards its neighbour) and ##T^\mu## is a vector tangent to the geodesic. The equation expresses the idea that the acceleration between two neighbouring geodesics is proportional to the curvature and, physically, it is the manifestation of gravitational tidal forces.
The six page document repeats the above and reviews covariant derivatives, parallel transport and geodesics as shown in the video, Riemann and Killing. It's at Commentary 3 Review chapter 3.pdf.
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