Curvature of a Riemannian Manifold | Riemannian Geometry
If you're interested in personal help, I've posted my tutoring services on Fiverr: https://www.fiverr.com/s/dDYkBlz
I have not had the opportunity to teach mathematics as much lately, given the amount of focus I have given to my research. I enjoy the process of teaching and interacting with students and so have decided to do some online tutoring.
In this lecture, we define the exponential mapping, the Riemannian curvature tensor, Ricci curvature tensor, and scalar curvature. The focus is on an intuitive explanation of the curvature tensors. The curvature tensor of a Riemannian metric is a very large stumbling block for many students in differential and Riemannian geometry. To those who understand the curvature tensor(s), however, this confusion seems strange as the curvature tensor(s) are very intuitive geometric objects.
We include a few examples of how to visualize curvature, using the fact that the (round metric on the) sphere has positive Riemannian curvature, the (flat metric on the) torus has vanishing curvature, and the (Poincaré metric on the) disk has negative Riemannian curvature.
It might be worth emphasizing that the curvature is a property of the Riemannian metric and not the underlying space (although the topology of the space may constrain the curvature). This will be explored in future videos on the subject.
👍 If you like the video, hit the like button and subscribe to the channel if you want to show your support/want to learn some mathematics.
🎸 Music in this video is from Epidemic Sound: https://www.epidemicsound.com/referra...
Ways to support the channel:
☕️ Support the growth of the channel by buying a coffee: https://www.buymeacoffee.com/kylebroder
🏆 Support the channel over on Patreon / kylebroder
FYI, these are affiliate links 💪🙏
These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.
Watch video Curvature of a Riemannian Manifold | Riemannian Geometry online without registration, duration hours minute second in high quality. This video was added by user Kyle Broder 02 February 2020, don't forget to share it with your friends and acquaintances, it has been viewed on our site 10,161 once and liked it 231 people.