Discussion Overview
The discussion revolves around the geometry that emerges from using a non-zero transverse vector field for projection in the context of a hypersurface in Euclidean space, as opposed to the standard Levi-Civita connection. Participants explore the implications of this approach on the induced geometry, including questions of compatibility and torsion.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant describes how a hypersurface obtains its Levi-Civita connection through orthogonal projection of vector fields and proposes using a non-zero transverse vector field instead.
- Another participant expresses the need for calculations to determine the nature of the geometry induced by the non-standard projection and requests a worked-out example.
- A different participant agrees to attempt constructing an example on the sphere and questions whether the resulting geometries would be torsion-free.
- One participant compares the proposed method to pulling back the Levi-Civita connection for a different metric on R^3, suggesting that the outcome may depend on the nature of the pullback metric.
- Another participant asserts confidence that the resulting geometry would be torsion-free.
Areas of Agreement / Disagreement
Participants express differing views on the implications of using a non-zero transverse vector field, with some questioning the compatibility of the induced geometry and others asserting it would be torsion-free. The discussion remains unresolved regarding the specific nature of the geometry that emerges.
Contextual Notes
Participants note the need for calculations and examples to clarify the implications of their ideas, indicating that assumptions about the nature of the vector field and the resulting geometry are not fully explored.