微分几何笔记讨论班的粗略总结Chp7-3

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7.4.5 Riemann curvature tensor with Levi-civita connection

We must remember the Christoffel symbols and it can be written out according to the metric.
In excise 7.10,
1. According to the distance, the metric elements are g_{rr}, g_{\theta\theta} and g_{\phi\phi};
2. Write out the Christoffel symbols,
3. At the k position, we have k=r,\theta,\phi, so we can divide the connection to three kinds of situation. During the calculation, the symmetry can be used to simplfy the work;
4. calculate the Riemann curvature tensor etc.
5. The second Bianchi identity has not be verified.
6. On page 270, contracting the indices has two ways, \delta^{\mu\nu} or g^{\mu\nu}, then we can get the einstein tersor.
7. The exercise 7.12 and 7.13 should be finished, and We will release it in a pdf docment.

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