Discussion Overview
The discussion centers around finding books on the applications of differential geometry beyond its traditional uses in relativity. Participants express interest in various fields where differential geometry might be applied, including classical mechanics, control theory, solid mechanics, and information theory. The conversation also touches on related topics such as fractional calculus and discrete differential geometry.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant, James, seeks recommendations for books on the applications of differential geometry, expressing a preference for works that include chapters on applications rather than solely theoretical texts.
- Another participant mentions a book by Schutz that discusses applications in thermodynamics, Hamiltonian mechanics, electromagnetism, fluids, and cosmology.
- Burke's "Applied Differential Geometry" is suggested as a resource for applications.
- Several references to works on discrete differential geometry and computational electromagnetism are provided, highlighting their relevance to the discussion.
- Participants discuss the concept of geometric mechanics, which applies differential geometry to classical mechanics, and mention various books related to this field.
- Information geometry is noted as an application of differential geometry, with some participants expressing a preference for stochastic geometry instead.
- One participant expresses interest in advanced mechanics and the application of manifold theory and tensors in dynamics.
- Global Calculus by S. Ramanan is suggested for its unique approach, although it is noted that it does not focus on applications.
- Discussion includes advanced topics such as sheaves, exact sequences, and cohomology, indicating a depth of interest in the mathematical foundations of differential geometry.
Areas of Agreement / Disagreement
Participants share various recommendations and express interest in different applications of differential geometry, but no consensus is reached on a single best resource or approach. Multiple competing views on the relevance of different applications and topics remain present.
Contextual Notes
Some participants express uncertainty about the specific applications of differential geometry in fields like aerodynamics and the extent to which discrete differential geometry can parallel traditional concepts. There is also a lack of clarity on the organization and coherence of some existing texts on discrete differential geometry.
Who May Find This Useful
Readers interested in the applications of differential geometry in physics, mathematics, and engineering, as well as those exploring advanced topics in mechanics and related mathematical concepts.
