Lecture summaries

Here are brief summaries of what was covered in lecture, with references to the text book.

8/30: Syllabus. Parametrized curves. The tangent vector. Arc-length. Examples. Reference: Textbook, Chapters 1.1- 1.2.

9/1: Periodic and closed curves. Self-intersections. Reparametrization. Unit-speed curves. Unit-speed curves are parametrized by arc-length. The acceleration vector and the tangent vector for a unit speed curve. Reference: Textbook, Chapters 1.3-1.4.

9/6: Regular curves. Regular curves can be re-parametrized to be unit speed. The curvature of a unit speed curve. The curvature of a regular curve (statement of formula). Examples. Reference: Textbook, Chapters 1.3, 2.1

9/8: The curvature of a regular curve (proof). The signed curvature and the turning angle. Examples. The total signed curvature of a closed curve is an integer multiple of 2 pi. Reference: Textbook, Chapters 1.3, 2.1-2.2.

9/13: Hopf's theorem (statement). The signed curvature determines a plane curve. The curvature does not determine a space curve. The torsion and the Frenet-Serret equations. The torsion and the curvature determine a space curve (statement). Reference: Textbook, Chapters 2.2-2.3, 3.1.

9/15: The torsion and the curvature determine a space curve (proof). The analogue of the Frenet-Serret equations in higher dimensions (will not be tested). Matrix exponentiation (will not be tested). Skew-symmetric matrices and orthogonal matrices. Simple curves. The Jordan Curve Theorem (statement). The Isoperimetric inequality (statement). The Four Vertex Theorem (statement). Reference: Textbook, Chapter 3, Chapter 2.3.

9/20: Proof of the Isoperimetric Inequality. Visualizing the torsion. Introduction to surfaces. Local patches and parametrization. Homeomorphisms and open sets. Reference: Textbook, Chapters 3.2, 4.1.

9/22: More about surfaces. Regular/smooth surfaces. Smooth maps. Reference: Textbook, Chapters 4.1 - 4.3

9/27: Tangents and derivatives. Normals and orientability. Compact surfaces. Reference: Textbook, Chapters 4.4 - 4.5, 5.4.

9/29: The first fundamental form. Examples. Lengths of curves on surfaces. Examples of surfaces. Reference: Textbook, 6.1, 5.1-5.3

10/4: Local isometries and local diffeomorphisms. Local isometries and the first fundamental form. Reference: Textbook, 6.2.

End of Midterm 1 lectures

10/6: More about local isometries. When is a local diffeomorphism a local isometry? Angles and conformal maps. Conformality and the first fundamental form. Statement of the Inverse Function Theorem. Reference: Textbook, 6.2-6.3, 5.6.

10/11: First midterm.

10/13: Applications of the Inverse Function Theorem. Reference: Textbook, 5.6.

10/18: Spherical geometry. The second fundamental form. Reference: Textbook, 6.5, 7.1.

10/20: The Gauss map and the Weingarten map. Normal and geodesic curvatures. Reference: Textbook, 7.2, 7.3.

10/25: Parallel vector fields. Covariant derivative. The Gauss equations. Reference: Textbook, 7.4.

10/27: Parallel transport. Computations in the case of the sphere. Reference: Textbook, 7.4.

11/1: The Gaussian and mean curvatures. The principal curvatures. Reference: Textbook, 8.1-8.2.

11/3: The principal curvatures maximize and minimize the normal curvature. Euler's Theorem. The Gaussian and mean curvatures in co-ordinates. Reference: Textbook, 8.1-8.2.

11/8: The Weingarten matrix in co-ordinates. Curvature, areas, and the Gauss map. Classification of surfaces with all points umbillic. The principal curvatures and the local form around a point. Reference: Textbook, 8.1-8.2.

11/10: Construction of a surface of constant negative curvature. Classification of surfaces of 0 curvature -- part 1, such surfaces are ruled. Surfaces of constant mean curvature and soap bubbles (statement). Reference: Textbook, 8.3-8.5.

End of Midterm 2 lectures

11/15: Classification of flat surfaces -- part 2. Compact surfaces have a point of positive curvature. Reference: Textbook, 8.4-8.6.

Midterm 2

11/22: More about points of positive curvature. Definition of geodesics and first properties. The geodesic equations and their significance. Refererence: Textbook, 8.6, 9.1.

11/29: Geodesics and local isometries. Geodesics on surfaces of revolution. Clairaut's Theorem. Classification of geodesics on the pseudosphere. Reference: Textbook, 9.2 - 9.3.

12/1: Geodesics from the variational point of view. Reference: Textbook, 9.4.

12/6: The Theorem Egregium and some of its consequences. Reference: Textbook, 10.1 - 10.3.