5/10/13: Last homework collected and discussed. [Text: Geodesics in
the Poincaré upper half-plane. Chapter 15 (through p. 292);
Lecture notes; do Carmo Riemannian Geometry, Chapters 2 and 4.]
5/10/13: Last homework collected and discussed. [Text: Geodesics in
the Poincaré upper half-plane. Chapter 15 (through p. 292);
Lecture notes; do Carmo Riemannian Geometry, Chapters 2 and 4.]
5/8/13: Tangent vectors, tangent space, vector and tensor
fields. Riemannian manifolds. Connections and covariant
derivatives. [Text: Lecture notes; do Carmo Riemannian
Geometry, Chapters 2 and 4.]
5/6/13: Smooth manifolds. [Text: Chapter 15 (through
p. 292).]
5/3/13: Discussion of Midterm Test II. [Text: Chapters 9, 10 (through
p. 195), 11 (through the top of p. 210), and the material covered in
class on constant-curvature surfaces, non-Euclidean geometries, the
hyperboloid, the Poincare disk and upper half-plane models of
hyperbolic geometry. (This roughly corresponds to pp. 218, 225, 228,
262-269 from Chapters 12 and 14 in the text, but the way we dealt with
this material was quite different and much simpler.)]
5/1/13: Midterm Test II. [Text: Chapters 9, 10 (through p. 195), 11
(through the top of p. 210), and the material covered in class on
constant-curvature surfaces, non-Euclidean geometries, the
hyperboloid, the Poincare disk and upper half-plane models of
hyperbolic geometry. (This roughly corresponds to pp. 218, 225, 228,
262-269 from Chapters 12 and 14 in the text, but the way we dealt with
this material was quite different and much simpler.)]
4/29/13 (Denis Bashkirov will be substituting substituting for me):
Discussion of the sample exam, continued. [Lecture notes only!]
4/26/13 (I will be out of town; Denis Bashkirov will be substituting
substituting for me): Review: discussion of the sample exam. [Lecture
notes only!]
4/24/13: Homework collected and discussed. The metric in
Poincaré model. The upper half-plane model of hyperbolic
geometry. The upper half-plane with the metric induced from the
hyperboloid in the pseudo-Euclidean space is locally isometric to the
Beltrami surface (the surface of revolution of the tractrix) and
therefore has Gaussian curvature -1. [Lecture notes only!]
4/22/13: Spherical and hyperbolic geometry. The stereographic
projection of the hyperboloid and the Poincaré model. [Lecture
notes only!]
4/19/13: Hyperbolic geometry preliminaries: the pseudo-Euclidean
spaces Rsn and pseudospheres; the
two-sheeted hyperboloid as a pseudosphere of imaginary radius; the
notions of a point and a line on the pseudosphere; the stereographic
projection to the Poincaré disk. [Lecture notes only!]
4/17/13: Homework collected. Positive, negative, and constant
curvature surfaces. The surface of revolution of a tractrix as a
pseudosphere. [Text: Chapter 11: 201-210; Chapter 12: 218, 225, 228]
4/15/13: Consequences of the Gauss-Bonnet theorem: torus cannot have a
constant Gaussian curvature; the absence of closed geodesics bounding
a simply conected region on a surface of nonpositive Gaussian
curvature; classification of Platonic solids. [Text: Chapter 11:
209-210, 217]
4/12/13: The Gauss-Bonnnnet theorem for closed surfaces. [Text: Chapter
11: 205-209]
4/10/13: Homework collected and discussed. The Gauss-Bonnet
theorem. [Text: Chapter 10: 192-195; Chapter 11: 201-205]
4/8/13: Hopf's Umlaufsatz. [Text: Chapter 11 through p. 203]
4/5/13: Geodesics: the distance minimizing property, the existence and
uniqueness, the exponential map. [Text: Chapter 10
through p. 195, Chapter 11 through p. 203]
4/3/13: Homework collected and discussed. [Text: Chapter 10: 188-191]
4/1/13: Geodesics and the geodesic equations. Clairaut's
relation. [Text: Chapter 10: 188-191]
3/29/13: The geodesic curvature and intrinsic normal for a curve on a
surface. [Text: Chapter 10: 185-187]
3/27/13: Homework collected and discussed. [Text: Chapter 9]
3/25/13: The fundamental theorem for surfaces. Special
coordinates. [Text: Chapter 9 (pp. 178-183)]
3/16-24/13: The Spring Break. But do not forget about the homework,
which is due on Wednesday after the break.
3/15/13: The Christoffel symbols. Theorema Egregium. The Gauss and
Mainardi-Codazzi equations. [Text: Chapter 9 (pp. 173-178)]
3/13/13: Homework collected and discussed. [Text: Chapters 8 and 9
(pp. 171-173)]
3/11/13: Graded exams handed out. Isometries. Intrinsic
geometry. [Text: Chapter 9 (pp. 171-173)]
3/8/13: Discussion of Midterm Test I. Minimal surfaces. [Text:
Chapters 5-8, skipping the Involutes and Evolutes section from
Chapter 5, the Appendix to Chapter 6, and Chapter
7bis.]
3/6/13: Midterm Test I. [Text: Chapters 5-8, skipping the Involutes
and Evolutes section from Chapter 5, the Appendix to Chapter 6,
and Chapter 7bis]
3/4/13: Discussion of Sample Test I. [Text: Chapters 5-8, skipping
the Involutes and Evolutes section from Chapter 5, the Appendix
to Chapter 6, and Chapter 7bis]
3/1/13: The Gaussian and mean curvatures. Formulas for the matrix of
the differential dNp of the Gauss map N through the
matrices of the first and second fundamental forms. [Text: Chapter 8
through the end]
2/27/13: Homework collected and discussed. The second fundamental
form. [Text: Chapter 8 through p. 164]
2/25/13: The differential of the Gauss map. Example of computing it
and the principal curvatures and directions. [Text: Chapter 8 through
p. 161]
2/22/13: Classification of points on a surface: elliptic, hyperbolic,
parabolic, planar, umbilic. The Gauss map. [Text: Chapter 8 through
p. 160]
2/20/13: Homework collected at the beginning of the class and
discussed. Euler's approach to curvature on a surface by taking
cross-sections by normal planes. Normal curvatures of a surface,
Euler's theorem, principal curvatures, principal directions. [Text:
Chapter 7, Chapter 8 through p. 158]
2/18/13: Lengths, angles, areas on a surface. [Text: Chapter 7 through
the end]
2/15/13: The first fundamental form. [Text: The corresponding section
from Chapter 7 (pp. 128-130)]
2/13/13: Homework collected at the beginning of the class and then
Problems 7.5 and H discussed. [Text: Chapter 7 through p. 128]
2/11/13: Surfaces and the tangent plane. [Text: Chapter 7 through
p. 128]
2/8/13: Au×Av = A(u×v) for A in SO(3). Introduction to
surfaces. Parametrizing the unit sphere. [Text: Chapter 7 through
Example (2) on p. 119]
2/6/13: Homework collected at the beginning of the class and then
discussed. [Text: Chapter 6 skipping the Appendix]
2/4/13: Space curves: computation of curvature and torsion, proof of
the Frenet-Serret formulas. The Fundamental theorem for space
curves. [Text: Chapter 6 before the Appendix]
2/1/13: The Fundamental theorem for plane curves. Space curves:
curvature and torsion, Frenet-Serret formulas. [Text: The Fundamental
theorem from Chapter 5, Chapter 6 through the wording of Theorem 6.5]
1/30/13: Homework collected at the beginning of the class and then
discussed. Correction about taking the second derivative of a
composite function α(t(s)) in the proof of the formula (Theorem
5.10) for curvature discussed. [Text: Chapter 5 from the Tractrix
section up to, but not including the Fundamental theorem]
1/28/13: Oriented curvature, tractrix. [Text: Chapter 5 from the
Tractrix section up to, but not including the Fundamental Theorem]
1/25/13: Arc length parametrization. Plane curves: osculating circle,
center of curvature, curvature. [Text: Chapter 5 after Proposition 5.4
up to, but not including the Tractrix section]
1/23/13: Introduction. Smooth and regular curves
in Rn, tangent vectors, speed, arc length,
reparametrization. [Syllabus. Text:
Chapter 5 through Proposition 5.4]
Last modified: (2013-05-13 12:45:05 CDT)