Source: https://math.berkeley.edu/~reshetik/140-2010.html
Timestamp: 2019-04-18 20:50:33+00:00

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This course is an introduction into metric differential geometry. It will start with the geometry of curves on a plane and in 3-dimensional Euclidean space. In this part of the course we will focus on Frenet formulae and the isoperimetric inequality. Then we will study surfaces in 3-dimensional Euclidean space. In this part of the course important subjects are first and second fundamental forms, Gaussian and mean curvatures, the notion of an isometry, geodesic, and the parallelism. Gauss-Bonnet theorem will be the next subject. If time permit, the last part of the course will be an introduction in higher dimensional Riemannian geometry.
R. Millman and G. Parker, Elements of Differential Geometry, Prentice Hall.
The will be two midterms in class (2/12, and 3/17). The final exam is on Tuesday 11, 8-11am, the final exam group is 5. The final grade will be assigned on the basis of the results of homework (15%), midterms (25% each), and of final exam (35%).
Below is a tentative schedule of lectures with some notes. It will expand as the course will progress.
Introduction, review of linear algebra in R^3, scalar product, vector product, its geometrical meaning, parametric descrciption of a line and a plane in R^3, description of planes and lines in R^3 by systems of linear equations.
Curves in 3-dimensional Euclidean plane. Curves can be defined by equations. They can be defined parametrically. Parametrized curves. Curve as a subset of R^3 is the image of the parametrization mapping. Tangen vectors. Reparametrization. The arclength. It does not depend on a parametrization. The arclength function.
Tangent vectors, normal vectors, curvature, and the torsion of a curve. Homework for material on Lectures 1-3 is due to Monday, Feb. 1. §1.4: 1cd, §1.5: 1, 2 §2.1: 8, 9 §2.2: 5, 8 §2.3: 2, 6, 7.
The Picard theorem, the Fundamental Theorem of Curves. Example of a helix.
Curvature of a plane curve, the rotation index, the formulation of the Rotation Index Theorem. Homework, due to Monday, Feb.8: §2.4: 1, 4, 5 (for 3.2), 10, 14; §2.5: 3, 7; §2.6: 3, 8 (this homework will be graded).
Convexity. Simple closed regular curve is convex if and onl if the curvature has constant sign. Isoperimetric inequality.
Review for the first midterm.
The discussion of problems from the first midterm.
Arclength. Metric and acrlength as intrinsic notions on a surface. Orientation of a surface.
Vector field along a curve. Vector field parallel along a curve. Parallel transport along a curve. Maximally straigh curves. Maximally straight=geodesic.
Second fundamental form. Weingarten map as a composition of the first and the second fundamental forms. Homework for next Monday, March 15 : � 4.7: 4, 7 � 4.8: 1, 2, 10.
Gauss map. Gauss theorem (Gauss curvature is the limit of areas). Hyperbolic, elliptic, parabolic, and flat points on a surface. Asymptotic directions.
Review for the second midterm.
Elements of linear algebra: tensor product of vector spaces, wedge product. Vector bundles.
Tangent space to a manifold. Tangent bundle.
Tangent bundle, vector fields, cotangent bundle, differential forms. Operations with differential forms.Recommended reading: Chapter 2 of John Lee's book.
Connections on vector bundles and linear connections. Chapter 3.
Levi-Civita connection. Properties. Chapter 4. Homework for Monday, April 12: 3-7, 3-6, 4-1, 4-5 from J. Lee book.
Properties of connections, Riemannian curvature.
Review of connectons, Levi-Civita connection, and Riemannian curvature.
Continuation of Riemannian curvature. Torsion of a connection.
End of the proof of Gauss-Bonnet formula. the Gauss-Bonnet theorem.
Review. Suggested problems: Millman and Parker: 1) p. 137: 8.3, 8.8, 8.11, 2)7.1, 7.3, 7.6, 7.7, 3)p.121, 6.2, 6.4, 4) Prove that all geodesics on a sphere are large circles.
A summer research program for undergraduates will take place this summer in Tucson. The topics will be on discrete geometry and geometric flows on Lie groups. I have attached a poster for the program, and the program's website is here. It is run by Dave Glickenstein and Andrea Young.
Differential Geometry by T. Shifrin.

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