Have been so entrenched in the mathematical foundation of science, it is sometimes nice to hear the pragmatic side of the story. Well this first lecture (second of the semester since I missed the first one last week) was quite mathematical in nature, at least to the physicist. It basically covered some tensor algebra as well as intuitive definition of Riemannian manifold (for which they used the more traditional term surface). A large portion of the lecture was devoted to explaining the concept of tensor transformation, which I had some understanding of before coming. The one confusion that was resolved for me, however, was the difference between index raising/lowering and change of coordinate. They look quite similar indeed.
The professor made a rather specific comment that n-manifolds can’t be imbedded in n+1 Euclidean (this word he never used) spaces in general. But someone in the audience was apparently aware of Nash’s result on Euclidean embedding of Riemannian manifold.
The first part of the lecture was quite enlightening: many new developments in cosmology and astrophysics was outlined, especially the definition of pulsar, double pulsar, Quasa, black hole, and the four predictions given by General Relativity. The theoretical value of SR and GR was made clear: SR came to rescue certain physical dilemmas arising at the end of the ninteenth century, whereas GR was mainly of aesthetic value to physicists.