SUMMARY
A periodic curve can still be considered an immersed submanifold of a manifold M. The curve map y, which maps an interval to M, is periodic and thus not injective. However, if y' is nonzero for all t, y qualifies as an immersion. The periodic nature allows the curve to be represented as a map Y from S^1 to M, where Y can be an injective immersion if y is injective during each period.
PREREQUISITES
- Understanding of immersed submanifolds in differential geometry
- Knowledge of periodic functions and their properties
- Familiarity with the concept of injective immersions
- Basic grasp of manifold theory
NEXT STEPS
- Study the properties of immersed submanifolds in differential geometry
- Learn about periodic maps and their implications in manifold theory
- Explore the relationship between injective immersions and periodic curves
- Investigate the topology of S^1 and its applications in manifold mappings
USEFUL FOR
Mathematicians, differential geometers, and students studying topology and manifold theory will benefit from this discussion.