N. Reshetikhin

office: 917 Evans Hall
email: reshetik at math.berkeley.edu
lectures: MWF 1:00-2:00pm, Room 3 Evans
office hours: Fridays 3:30-4:30pm

Tentative syllabus

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.

Textbook

R. Millman and G. Parker, Elements of Differential Geometry, Prentice Hall.

Exams

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%).

Lectures

Below is a tentative schedule of lectures with some notes. It will expand as the course will progress.

Lecture 1, W. Jan. 20,

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.

Lecture 2, F. Jan. 22

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.

Lecture 3, M. Jan. 25

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.

Lecture 4, W. Jan. 27

Torsion, curvature, Frenet-Serret theorem.

Lecture 5, F. Jan. 29

The Picard theorem, the Fundamental Theorem of Curves. Example of a helix.

Lecture 6, M. Feb. 1

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).

Lecture 7, W. Feb. 3

The proof of the Rotation index theorem. The discussion of parametrization of curves and the notion of a manifold on the example of a 1-dimensional manifold. This homework is due Wednesday, Feb. 17. §3.1: 3; §3.2: 1, 2, 3; §2.4: 11, 13; § 2.5: 4; § 2.6: 7; § 3.3: 2; § 3.4: 1, 2; Solutions to homework 1

Lecture 8, F. Feb. 5

Convexity. Simple closed regular curve is convex if and onl if the curvature has constant sign. Isoperimetric inequality.

Lecture 9, M. Feb. 8

Isoperimetric inequality, 4-point theorem.

Lecture 10, W. Feb. 10,

Review for the first midterm.

Lecture 11, F. Feb. 12

First midterm.

Holiday, W. Feb. 15

Lecture 12, W. Feb. 17

Manifolds are defined. Surfaces, as 2-dimensional manifolds, embedded surfaces. Homework due to Monday, Feb. 22: 4.1: 1, 2, 10, 5, 9 4.2: 1, 2, 4. Solutions to homework 2

Lecture 13, F. Feb. 19

The discussion of problems from the first midterm.

Lecture 14, M. Feb. 22

The notion of a tangent plane to a surface. Tangent space. Homework due next Friday, March : 4.3: 1, 7 4.4: 2, 4, 5

Lecture 15, W. Feb. 24

Metric: first fundamental form. Change of coordinates. Tensor. Metric is as a tensor. Solutions to homework 3

Lecture 16, F. Feb. 26

Arclength. Metric and acrlength as intrinsic notions on a surface. Orientation of a surface.

Lecture 17, M. March 1

Normal and geodesic curvatures of a curve on a surface. Christoffel symbols. Homework, due to Monday, March 8: 4.5: 5.6, 5.10, 4.6: 3, 4,

Lecture 18, W. March 3

Christoffel symbols. Geodesic curvature is intrinsic. Geodesic curve. Solutions to homework 4

Lecture 19, F. March 5

Vector field along a curve. Vector field parallel along a curve. Parallel transport along a curve. Maximally straigh curves. Maximally straight=geodesic.

Lecture 20, M. March 8

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.

Lecture 21, W. March 10

Principal curvatures of a surface. Principal directions. Gauss and mean curvatures. Solutions to homework 5

Lecture 22, F. March 12

Gauss map. Gauss theorem (Gauss curvature is the limit of areas). Hyperbolic, elliptic, parabolic, and flat points on a surface. Asymptotic directions.

Lecture 23, M. March 15

Review for the second midterm.

Lecture 24, W. March 17

Second midterm.

Lecture 25, F. March 19

Homework.

Spring recess, M. March 22

No class

Spring recess, W. March 24

No class

Spring recess, F. March 26

No class

Lecture 26, M. March 29

Elements of linear algebra: tensor product of vector spaces, wedge product. Vector bundles.

Lecture 27, W. March 31

Tangent space to a manifold. Tangent bundle.

Lecture 28, F. April 2

Tangent bundle, vector fields, cotangent bundle, differential forms. Operations with differential forms.Recommended reading: Chapter 2 of John Lee's book.

Lecture 29, M. April 5

Connections on vector bundles and linear connections. Chapter 3.

Lecture 30, W. April 7

Levi-Civita connection. Properties. Chapter 4. Homework for Monday, April 12: 3-7, 3-6, 4-1, 4-5 from J. Lee book.

Lecture 31, F. April 9

Properties of connections, Riemannian curvature.

Lecture 32, M. April 12

Review of connectons, Levi-Civita connection, and Riemannian curvature.

Lecture 33, W. April 14

Review of homework, examples.

Lecture 34, F. April 16

Continuation of Riemannian curvature. Torsion of a connection.

Lecture 35, M. April 19

Integration of forms.

Lecture 36, W. April 21

Riemannian volume form.

Lecture 37, F. April 23

The Stokes theorem.

Lecture 38, M. April 26

The Gauss-Bonnet formula.

Lecture 39, W. April 28

End of the proof of Gauss-Bonnet formula. the Gauss-Bonnet theorem.

Lecture 40, F. April 30

Review

Review session 1, May 3

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.

Review session 2, May 7

Review
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