Elementary Differential Geometry
James Cook's Elementary Differential Geometry Homepage

The initial big picture:

Elementary differential geometry is centered around problems of curves and surfaces in three dimensional euclidean space. We're using Barret Oneil's excellent text this semester.
Oneil uses linear algebra and differential forms throughout his text. I am excited about learning the method of moving frames for surfaces in 3-space. Ideally, I'd like to start reading some papers which generalize this method to surfaces in more exotic three dimensional manifolds. If our course goes very well, perhaps I'll be able to show you something about that other world of ideas at the end.
This course is the natural bridge to abstract manifold theory. I will avoid the temptation to generalize our work as we go. In some sense, advanced calculus is a more abstract course as we insist on treating many questions in n-dimensions. Here, in this course, usually n=3.

Guide to Self-Paced Differential Geometry Course:
In Summer 2015 I wrote these notes:

• Elementary Differential Geometry:

• from which I gave the Lectures based on O'neill, Kuhnel for Test 1. Then for Test 2 I simply recycled my old course notes plus a few new hand-written pages for Chapter 4. Then I talked through my notes from Tapp to help build-up to the final exam project. I have links to videos which are not absolutely essential to watch, I think you could get enough from the texts and the notes alone. But, perhaps the videos help. I also included scans of what I wrote in landscape for the Lectures.
• Lecture 1: [ notes from video ] video (5:23)
overview, goals, prerequisites:
• Lecture 2: [ notes from video ] part 1 (23:30) part 2 (39:18)
(Section 1.1-1.3 of Oneill and Chapter 1 of Kuhnel): basic notations of points and vectors, tangent spaces, directional derviatives, derivation notation for vectors.
• Lecture 3: [ notes from video ] part 1 (39:42) part 2 (12:21) part 3 (2:56)
(Section 1.5-1.6 of Oneill): differential forms in Euclidean three dimensional space. Wedge product and exterior differentiation as you might see at the end of an ambitious multivariate calculus course. Kuhnel introduces this topic much later at a higher level of sophistication.
• Lecture 4: [ notes from video ] part 1 (32:59) part 2 (36:28)
(Section 1.4 and 1.7 of Oneill and Chapter 1 of Kuhnel): curves and velocity and an operator, major theorems and terms from advanced calculus. Component functions, derivative of vector-valued functions of several variables. Here I will be a bit more explicit about derivations than Oneill.
• Lecture 5: [ notes from video ] part 1 (53:32) part 2 (46:27)
(Sections 2.1-2.3 of Oneill): tangent space as an inner product space, curves in three dimensional space, velocity and acceleration (or first and second derivatives of parametrized curve if you prefer)
• Lecture 6: [ notes from video ] part 1 (57:01) part 2 (20:08)
(Section 2.3- 2.4 of Oneill): Frenet frame and the Frenet formulas, non-unit speed curves, further examples of curves in three dimensions.
• Lecture 7: [ notes from video ] part 1 (31:07) part 2 (17:49) part 3 (32:25)
(Section 2.5-2.7 of Oneill): covariant derivative, frame fields and connection form introduced. This is the first showcase of the frame field technique. We also prove some essential calculus for matrices of forms in Part 3.
• Lecture 8: [ notes from video ] video (59:51)
(Section 2.8): the structural equations. Here we use matrix-valued differential forms to understand the geometry of frame fields. This section is important as it is later specialized to surfaces.
• Lecture 9: [ notes from video ] part 1 (25:00) part 2 (28:30)
(Sections 3.1-3.3 of Oneill): isometries and orientation in Euclidean space.
• Lecture 10: [ notes from video ] video (34:36)
(Chapter 2 of Kuhnel): theory of Frenet curves generalized to curves in n-dimensional Euclidean space. Naturally, in n=3 case this is redundant, but, I think we�ll gain much deeper understanding of Oneill from this perspective. On the other hand, Kuhnel does not discuss covariant derivatives here as the approach to surfaces in Kuhnel does not use Cartan�s equations as primary.
• Lecture 11: [ notes from video ] part 1 (27:28) part 2 (51:12)
(Sections 3.4-3.5 of Oneill): Euclidean geometry and congruence of curves.

• [ MISSION 1:] in preparation for Test 1 (also should do the recommended problems for which solutions are posted towards base of this page).
• TEST 1: covers Chapters 1-3 of Oneil and Chapters 1 and 2 of Kuhnel with focus on curves and frames.

• Lecture 12: [ notes from video ] part 1 (10:59) part 2 (19:02) part 3 (8:37) part 4 (16:17) part 5 (49:12)
(Sections 4.1-4.3 of Oneill): surfaces in three dimensions, patches, examples, differentiable functions and tangent vectors to a surface.
• Lecture 13: [ notes from video ] part 1 (40:22) part 2 (5:18) part 3 (41:32) part 4 (10:48) part 5 (19:07) part 6 (11:15)
(Sections 4.4-4.6 of Oneill): differential forms, mappings of surfaces, integration of forms on surfaces.
• Lecture 14: [ notes from video ] part 1 (43:02) part 2 (15:24)
(Sections 4.7-4.8 of Oneill): topological concepts on surfaces, manifolds.
• Lecture 15: [ notes from video ] part 1 (41:04) part 2 (35:04) part 3 (38:49)
(Sections 5.1-5.3 of Oneill): shape operator, normal and Gaussian curvature.
• Lecture 16: [ notes from video ] video (35:01)
(Sections 5.4-5.5 of Oneill): alphabet soup of classical formulas E,F,G, L,M,N and also the formulas for level surfaces, computational techniques.
• Lecture 17: [ notes from video ] video (56:21)
(Sections 5.6 - 5.7 of Oneill): special curves and surfaces of revolution.
• Lecture 18: [ notes from video ] video (41:46)
(Sections 6.1-6.2 of Oneill): frame fields adapted to surfaces, fundamental structural equations, form calculations.
• Lecture 19: [ notes from video ] video (13:31)
(Section 6.3 of Oneill): some global theorems about vanishing curvature, planes, spheres, compact and umbilic surfaces.
• Lecture 20: [ notes from video ] part 1 (26:30) part 2 (23:34)
(Sections 6.4-6.5 of Oneill): local isometries and the intrinsic geometry of surfaces. Gauss� theorema egregium as argued in the language of differential forms.
• Lecture 21: [ notes from video ] part 1 (43:02) part 2 (13:01)
(Sections 6.6-6.8 of Oneill): orthogonal coordinates give nice formulas, orientation and integration, total curvature. (the total curvature on its own is perhaps not so exciting, but, when you see how it appears in the Gauss-Bonnet Theorem in 7.6.4 you�ll appreciate studying it here)
• Lecture 22: [ notes from video ] video (6:32)
(Section 6.9 of Oneill): congruence of surfaces, here we see that having the same shape operator amounts to having the same shape in the sense that one surface is related to the other by a rigid motion. This generalizes Lecture 10.
• Lecture 23: [ notes from video ] video (26:57)
(Section 7.1 of Oneill): concept of an abstract geometric surface.
• Lecture 24: [ notes from video ] video (42:03)
(Section 7.2 of Oneill): Gaussian curvature developed in the abstract case.
• Lecture 25: [ notes from video ] video (41:16)
(Section 7.3 of Oneill): covariant derivatives motivated and discussed for an abstract geometric surface.
• Lecture 26: [ notes from video ] video (29:56)
(Sections 7.4-7.5 of Oneill): geodesics and Clairaut parametrizations.
• Lecture 27: [ notes from video ] part 1 (55:54) part 2 (3:42)
(Section 7.6 of Oneill): Gauss Bonnet Theorem
• Lecture 28: [ notes from video ] video (29:56)
(Section 7.7 of Oneill): applications of Gauss Bonnet.
• Lecture 29: [ notes from video ] video (29:56)
(Section 3A-3B of Kuhnel): the I, II and III forms on a surface.
• Lecture 30: (Section 3C of Kuhnel): surfaces of revolution and ruled surfaces (not given)
• Lecture 31: (Section 3D of Kuhnel): minimal surfaces (not given)

• [ MISSION 2:] in preparation for Test 2 (also should do the recommended problems for which solutions are posted towards base of this page).
• TEST 2 on Chapters 4-7 of Oneill, that is, on surfaces.

• Lecture 32: (Sections 5A-5B of Kuhnel): n-dimensional manifold and tangent space (not yet given)
• Lecture 33: (Section 5C of Kuhnel): Riemannian metrics (not yet given)
• Lecture 34: (Section 5D of Kuhnel): Riemannian connections (not yet given)
• Lecture 35: (Section 6A of Kuhnel): tensors (not yet given)
• Lecture 36: (Section 6B of Kuhnel): sectional curvature (not yet given)
• Lecture 37: (Section 6C of Kuhnel): Einstein and Ricci tensors (not yet given)
• Lecture 38-46 relabled under their own title of "crash course in matrix algebras" : on my You Tube Channel where I posted 14 little talks which cover the main results from Tapp's text:
• Part 1: (13:41)
In this talk we cover pages 1 to 4 of my notes where the K-notation for K=R,C,H is explained and the general linear group of nxn matrices over K is defined. We also begin to explain mapping which embeds nxn complex matrices in 2nx2n real matrices.
• Part 2: (20:16)
Pages 4 to 6 of my notes. Here we study the complex matrices represented as real matrices and quaternionic matrices represented as complex matrices. These natural homomorphisms allow us to reduce everything to a problem of sufficiently large real matrices. Conversely, for a set of real matrices which satisfies the right conditions with respect to the complex or quaternionic structure we can trade the real problem for a corresponding complex or quaternionic problem.
• Part 3: (15:25)
Based on pages 7 to 9 of my notes. Inner product for n-tuples over K=R,C, H are described using appropriate conjugations. Also, the isometries of these standard inner products naturally give rise to On(K) which produces at once the seemingly distinct matrix groups O(n) over R, U(n) over C and Sp(n) over H; that is orthogonal, unitary and symplectic matrices
• Part 4: (13:47)
Covers pages 10 to 12 of my notes. Calculus on matrix groups exposed. In particular, we examine how curves of matrices through the identity matrix are used to define the tangent space to the identity which we find to have a Lie Algebra structure.
• Part 5: (16:57)
Covers page 13 to 15 of my notes. We characterize the Lie algebras of several standard examples by studying the velocity of matrix-curves through the identity. Then, we spend a few minutes deriving a beautiful formula through the manifest multilinearity of the levi-civita formula for the determinant. One of my favorite calculations.
• Part 6: (13:33)
Covers pages 16 to 19 of my notes. Here we sketch how transformations on Kn can be viewed as vector fields on Kn as points correspond naturally to vectors in the vector space Kn. Probably could use a lot more pictures here. Lie algebra of SO(n) proves a bit more challenging to derive than our previous examples.
• Part 7: (13:47)
Covers pages 20 to 21 of my notes. We study the matrix exponential which gives us a map between the Lie algebra and the Lie group.
• Part 8: (5:18)
Covers page 22 to 23 of my notes. We use the matrix exponential to generate the one-parameter groups for the matrix Lie group.
• Part 9: (21:27)
Covers pages 23 to 27 of my notes. The big adjoint is introduced as the derivative of the conjugate (in the group-theoretic sense of the term). Then we derive a few basic calculational tools which allow us to see the fascinating theorem about morphisms of Lie Groups transferring naturally to the corresponding Lie Algebras. Once more, the matrix exponential serves a deep and meaningful role in all this.
• Part 10: (7:08)
Covers pages 28 to 29 of my notes.We give an example of Lie algebras which are seemingly distinct, yet, share the same bracket-structure (hence are isomorphic). However, we caution the isomorphism of groups brings global topological questions to bear of which we only touch on here... Finally the homomorphism Ad is introduced to provide a homomorphism of the Lie group G on sets of invertible matrices of size dxd.
• Part 11: (9:11)
Covers pages 30 to 32 of my notes. Here we study yet more about the adjoint map, an interesting identity connecting the Lie algebra with the exponential of the adjoint. There is more to read in Tapp, I didn't touch on the Baker-Cambell-Hausdorff relation etc.
• Part 12: (7:54)
Covers page 33 of my notes. Lie correspondence theorem given and the concept of Spin(n) is briefly described.
• Part 13: (25:31)
Covers pages 34 to 38 of my notes. Higher dimensional torus is described. The maximal torus of a matrix group is defined and a standard presentation of the torus is given for each of the standard examples. A theorem about the center of the group is studied with the help of the maximal tori. We also describe how conjugation (in the group theoretic sense) moves us from the standard torus to other points in the group. We mention certain parts of the proof which are interesting to the study of e-vectors and and diagonalization of symmetric matrices in linear algebra.
• Part 14: (11:00)
Covers pages 39 to 40 of my notes. Lie algebra of the tori are detailed. Further detail on the conjugates of tori are also given as to place regular elements on just one such torus. Finally, the classification of compact matrix groups is given. We explain that all there is to find is direct products of SO(n), SU(n), Sp(n) and the 5 exceptional Lie groups. Of course, these results are due to an entirely different course of study. See Erdmann and Wildon's "Introduction to Lie Algebras" for a treatment which is essentially at the same level as Tapp.
• Final Exam Project.
Rough Notes from my Reading in Oneil:
• Chapter 1: preliminary notes on Chapter 1 (the next set has more of exterior derivatives and wedges to help those who are rusty and/or have never seen them).
• Chapter 1: summary notes on Chapter 1.
• Chapter 2 (2.1-2.4): notes on vector fields in R3 and the Frenet-Serret equations.
• Chapter 2 (2.5-2.7): notes on the covariant derivative in R3, frame fields and the reformulation of the covariant derivative in terms of the connection form of a frame field.
• Chapter 2 (2.8): notes on the structure equations of Cartan. In this section we begin to see how matrix-valued differential forms are used to calculate geometric data.
• Chapter 3 (3.1-3.3): notes on isometries of three dimensional space and their tangent maps.
• Chapter 4: calculus on surfaces in R3 ( unfinished, beware some typos on last couple pages).
• Chapter 5: shape operators and some classic formulas of Gauss
• Chapter 6: on the intrinsic geometry of surfaces via form calculus.
• Chapter 6: notes on integration and total curvature again for surfaces in euclidean 3-space
• Chapter 7: on geometric surfaces: how to create non-euclidean metrics on plane, pullback metrics, flat spheres and curved planes. Gauss's awesome theorem made a definition.
• Chapter 7: covariant derivatives, geodesic curves, intrinsic angles and geodesic curvatures.
Homework:
As I mentioned, my personal goal is to solve most the problems. However, that seems like a bit much for the class. Missions are given below:

• Mission 1: homework paired with Test 1 based on my lecture notes, O'neill and perhaps Kuhnel.

• Mission 2: homework paired with Test 2 based on my lecture notes, O'neill and perhaps Kuhnel.

• Missions 1-6 detailed I will update this from time to time (it lists the first few Missions as well as my weekly topic targets)

• I'll probably just let you look at my global solutions for this class. At a minimum, the solutions below should include most of the problems assigned in the Missions.

• note to self, delete this later: final exam solution.

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