- #31
RockyMarciano
- 588
- 43
Very benevolent of you.lavinia said:
Curvature of Flat Lorentz manifolds
- Context: Graduate
- Thread starter lavinia
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SUMMARY
Flat Lorentz manifolds, which are quotients of Minkowski space by a properly discontinuous group of Lorentz isometries, do not admit a flat Riemannian metric. This conclusion arises from the fact that such groups are not subgroups of the rigid motions of Euclidean space, leading to non-zero Riemannian curvature for any Levi-Civita connection. The discussion highlights the complexity of defining Minkowski space in mathematical contexts and its implications in particle physics, particularly regarding the curvature of Riemannian metrics on these manifolds.
PREREQUISITES- Understanding of Minkowski space and its properties
- Familiarity with Riemannian geometry and curvature concepts
- Knowledge of Lorentz isometries and their mathematical implications
- Basic principles of differential geometry and metric connections
- Research the properties of Minkowski space as a flat affine 4-manifold
- Explore the implications of Levi-Civita connections in Riemannian geometry
- Study the structure of flat Lorentz manifolds and their fundamental groups
- Investigate the relationship between Lorentz metrics and Riemannian metrics in theoretical physics
Mathematicians, physicists, and researchers interested in the geometry of spacetime, particularly those focusing on the implications of flat Lorentz manifolds in both mathematical and physical contexts.
- #32
RockyMarciano
- 588
- 43
@lavinia, would Minkowski space itself, for the timelike vector case(where the Lorentz group acts on the interior of the future timelike oriented cone) not have the same issue with its Lorentz isometries acting properly discontinuously and not being a subgroup of rigid euclidean motions?