Differential Geometry Lecture Notes

Introduction

• Lecturer: Claudia Reso
• Course: Differential Geometry
• No prior knowledge required
• Communication: Open office hours, no fixed schedule
• No provided notes; recommended book: The Geometry of Curves and Surfaces by Manfredo do Carmo
• Homework every two weeks (not graded), 5-6 exercises, important for learning

Overview of Differential Geometry

• Studies curvature and its effects on geometry and differential equations
• Focus on non-flat (curved) spaces
• Starts with one-dimensional objects (curves), then moves to two-dimensional (surfaces)

Curves

Definitions and Concepts

• Differentiable Curve: A map ( \alpha: I \to \mathbb{R}^3 )
  • ( I ): Interval, differentiable means ( C^\infty )
  • Tangent Vector: First derivative ( \alpha'(t) ) at ( t )
  • Planar Curve: Curve contained in a plane
  • Non-injective Curves: Can have self-intersections
  • Curves can have corners even if components are smooth

Examples

1. Straight Line: Defined by a point and direction vector
2. Circle: Parameterized by center and radius
3. Helix: Has linear Z component and circular XY projection

Length of Curves

• Defined using partitions and limits
• Use of integrals: ( \int_a^b ||\alpha'(t)|| , dt )
• Length is invariant under isometries (translations and rotations)

Parametrization

• Different parameterizations of the same curve can exist
• Reparametrization: Changing parameters does not change curve's length
• Diffeomorphism: Smooth map with smooth inverse
  • Example: ( \beta = \alpha \circ \phi )

Key Theorem and Proof

• Length remains constant under reparametrization

Looking Forward

• Explore if there is a canonical parametrization for curves
• Challenge arises in higher dimensions (surfaces) where canonical parametrization may not exist

Final Notes

• Understanding the basics of curves lays the groundwork for higher-dimensional geometry
• Prepare for next lecture focused on canonical parametrization of curves and its limitations in higher dimensions