**Course: **MATH 2971T

**Title: **Introduction to Differential Geometry

**Section: **100 (Class Number 14883) (Tutorial for students in the Honors Tutorial College, HTC)

**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Academic Year: **2019 - 2020

**Term: **Fall Semester

**Instructor: **Mark Barsamian

**Contact Information: **My contact information is posted on my web page.

**Office Hours for 2018 - 2019 Fall Semester: **8:45am - 9:30am Mon - Fri in Morton 538

**Course Description: **Curves in \( \mathbb{R}^{2} \) and \( \mathbb{R}^{3} \); Surfaces in \( \mathbb{R}^{3} \); Curvature; Geodesics; the Gauss-Bonnet Theorem

**Textbook Information: **

**Title: ***Elementary Differential Geometry, 2 ^{nd} Edition*

**Authors **Andrew Pressley

**Publisher: **Springer, 2012

**ISBN: **978-1-84882-890-2

**Calendar: **

**Reading:**Pressley Chapter 1: Curves in \( \mathbb{R}^{2} \) and \( \mathbb{R}^{3} \)- 1.1: What is a curve? Cartesian -vs- Parametrized presentation of curves. Tangent vector to a curve.
- 1.2: Arc Length, speed
- 1.3: Reparametrizations, Regular Curves
- 1.4: Closed Curves,
*T*-Periodic Curves

**Homework:**H01 Due Tue Sep 3- 1.1 # 3,4,7,8,9
- 1.2 # 1,2,3,4
- 1.3 # 1,2,3
- 1.4 # 1,2,3,4,5

**Reading:**Pressley Chapter 2: How much does a curve curve?- 2.1: Curvature
- 2.2: Plane Curves, Signed Curvature, Osculating Circle
- 2.3: Space Curves, Principal Normal Vector, Binormal Vector, Torsion, Frenet-Serret Equations
- 2.1: Curvature

**Homework:**H02 Due Wed Sep 11- 2.1 # 1,2
- 2.2 # 1,2,6,7
- 2.3 # 1,2,3,4,5 (typo in #4: should be \( \frac{d}{dt} \))

**Reading:**Pressley Chapter 3: Global Properties of [Plane] Curves- 3.1: Simple Closed Curves
- 3.2: The Isoperimetric Inequality
- 3.3: The Four Vertex Theorem

**Homework:**H03 Due Mon Sep 16- 3.1 # 1
- 3.2 # 2
- 3.3 # 2,3

**Exam 1:**Take-home exam covering Chapters 1 - 3, Assigned Tue Sep 17 at noon; Due Thu Sep 19 at noon**Reading:**Pressley Chapter 4: Surfaces in \( \mathbb{R}^{3} \)- 4.1: What is a Surface? Surface Patch, Atlas, Transition Map
- 4.2: Smooth Surfaces
- 4.3: Smooth Maps Between Surfaces, Smooth Functions from a Surface to \( \mathbb{R} \)

**Homework:**H04 Due Mon Sep 23- 4.1 # 2,3,4
- 4.2 # 2,3,5,6
- 4.3 # 1,2

**Reading:**Pressley Chapter 4: Surfaces in \( \mathbb{R}^{3} \)- 4.4: Tangent Vector to a Surface, Tangent Plane, Derivative of a Map Between Surfaces

**Homework:**H05 Due Fri Sep 27- Revisit 4.2 #6. Write an outline for the author's solution. (Just write an outline.)
- 4.4 # 1,2,3 (Be sure to include outline-type headings in your solutions.)
- In 4.4#2, note that the book is often casual about reparametrizations. We need to be more precise. A
*reparametrization of*\( \sigma \) is a map \( \tilde{\sigma} \) that can be expressed as \( \tilde{\sigma} = \sigma \circ \phi \) where \( \phi : \tilde{U} \rightarrow U \)

- In 4.4#2, note that the book is often casual about reparametrizations. We need to be more precise. A

**Reading:**Pressley Chapter 5: Examples of Surfaces- 4.5: Normals and Orientability, Oriented Surface
- 5.1 Level Surfaces

**Homework:**H06 Due Mon Oct 7- 4.5 # 1,2
- 5.1 # 2,3

**Reading:**Pressley Chapter 6: The First Fundamental Form- 6.1 Lengths of Curves on Surfaces

**Homework: H07**Due Fri Oct 11- 6.1 # 1,2,3,4

**Reading:**Pressley Chapter 6: The First Fundamental Form- 6.2 Isometries of Surfaces
- 6.3 Conformal Mappings of Surfaces

**Homework:**H08 Due Fri Oct 18- 6.2 # 1,2,3
- 6.3 # 1,2,4,5,6,7

**Reading:**Review Chapters 4,5,6**Homework:**None**Exam 2:**Take-home exam covering Chapters 4,5,6, Assigned Thu Oct 24 at 3pm; Due Mon Oct 28 at noon

**Reading:**Pressley Chapter 7: Curvature of Surfaces- 7.1 The Second Fundamental Form
- 7.2 The Gauss and Weingarten Maps
- 7.3 Normal and Geodesic Curvatures
- 7.4 Parallel Transport and the Covariant Derivative

**Homework:**H09 Due Mon Nov 4- 7.1 #1,2,3,4
- 7.2 #1,2,3
- 7.3 #1,2,3,4
- 7.4 #1

**Reading:**Pressley Chapter 8: Gaussian, mean, and principal curvatures- 8.1 Gaussian and mean curvatures

**Homework:**H10 Due Fri Nov 8- 8.1 #1,2,4,5,7

**Reading:**Pressley Chapter 8: Gaussian, mean, and principal curvatures- 8.2 Principal curvatures of a surface
- 8.3 Surfaces of constant Gaussian curvature
- 8.4 Flat Surfaces
- 8.5 Surfaces of constant mean curvature

**Homework:**H11 Due Fri Nov 15- 8.2 #1,2,3,4
- 8.4 #1
- 8.5 #1,2

**Reading:**Pressley Chapter 9: Geodesics- 9.1 Definition and basic properties
- 9.2 Geodesic equations
- 9.3 Geodesics on surfaces of revolution
- 9.4 Geodesics as shortest paths

**Homework:**H12 Due Mon Nov 25- 9.1 #1,2,5
- 9.2 #1,2,3,4,6
- 9.3 #1,2
- 9.4 #1,2

**Reading:**Pressley Section 9.5 Geodesic Coordinates and Section 10.2 Gauss's Theorem**Exam 3:**Take-home exam covering Chapters 7,8,9, Assigned Tue Nov 26 at 3pm; Due Thu, Dec 5 at 9am.

**Reading:**Chapter 13 (The Gauss-Bonnet theorem)**Final Exam:**Take-home exam covering Chapter 13, Assigned Thu Dec 5 at 3pm; Due Thu Dec 12 at 9am.

**Grading: **

During the semester, you will accumulate a * Points Total* of up to

**Homework:**Twelve @ 20 points each = 240 points possible**Exams:**- 2 exams @ 200 points each = 400 points possible
- 1 exam @ 160 points each = 160 points possible

**Final Exam:**200 points possible

At the end of the semester, your * Points Total* will be converted into your

- 900 - 1000 points = 90% - 100% = A-, A = You mastered all concepts, with no significant gaps.
- 800 - 899 points = 80% - 89.9% = B-, B, B+ = You mastered all essential concepts and many advanced concepts, but have some significant gaps.
- 700 - 799 points = 70% - 79.9% = C-, C, C+ = You mastered most essential concepts and some advanced concepts, but have many significant gaps.
- 600 - 699 points = 60% - 69.9% = D-, D, D+ = You mastered some essential concepts.
- 0 - 599 points = 0% - 59.9% = F = You did not master essential concepts..

**There is no curve.**

**Throughout the semester, your current scores and current course grade will be available in an online gradebook on the Blackboard system.**

**Homework: **

- Homework H01 Due Tue Sep 3
- 1.1 # 3,4,7,8,9 (note typo in book solutions for #9)
- 1.2 # 1,2,3,4
- 1.3 # 1,2,3
- 1.4 # 1,2,3,4,5

- Homework H02 Due Wed Sep 11
- 2.1 # 1,2
- 2.2 # 1,2,4,6
- 2.3 # 1,2,3,4,5 (typo in #4: should be \( \frac{d}{dt} \))

- Homework H03 Due Mon Sep 16
- 3.1 # 1
- 3.2 # 2
- 3.3 # 2,3

- Homework H04 Due Mon Sep 23
- 4.1 # 2,3,4
- 4.2 # 2,3,5,6
- 4.3 # 1,2

- Homework H05 Due Fri Sep 27
- Revisit 4.2 #6. Write an outline for the author's solution. (Just write an outline.)
- 4.4 # 1,2,3 (Be sure to include outline-type headings in your solutions.)
- In 4.4#2, note that the book is often casual about reparametrizations. We need to be more precise. A
*reparametrization of*\( \sigma \) is a map \( \tilde{\sigma} \) that can be expressed as \( \tilde{\sigma} = \sigma \circ \phi \) where \( \phi : \tilde{U} \rightarrow U \)

- In 4.4#2, note that the book is often casual about reparametrizations. We need to be more precise. A

- Homework H06 Due Mon Oct 7
- 4.5 # 1,2
- 5.1 # 2,3

- Homework H07 Due Fri Oct 11
- 6.1 # 1,2,3,4

- Homework H08 Due Fri Oct 18
- 6.2 # 1,2,3
- 6.3 # 1,2,4,5,6,7

- Homework H09 Due Mon Nov 4
- 7.1 #1,2,3,4
- 7.2 #1,2,3
- 7.3 #1,2,3,4
- 7.4 #1

- Homework H10 Due Fri Nov 8
- 8.1 #1,2,4,5,7

- Homework H11 Due Fri Nov 15
- 8.2 #1,2,3,4
- 8.4 #1
- 8.5 #1,2

- Homework H12 Due Mon Nov 25
- 9.1 #1,2,5
- 9.2 #1,2,3,4,6
- 9.3 #1,2
- 9.4 #1,2

- Homework H13 Due Fri Dec 6
- Homework Details TBA

page maintained by Mark Barsamian, last updated Nov 21, 2019