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# System prepended metadata

title: Differential Geometry
tags: [teaching]

---

---
title: Differential Geometry
tags: teaching
---

# Differential Geometry

## Unit 1: Review (2 weeks)
:page_with_curl: 2 homeworks

:::info
**Linear algebra**: Subspaces, inner products, cross products, orientations
- Week 1, [time=Jan 30 and Feb 1]

**Read:** Sections 1-1 and 1-3.
> may be that _general_ inner products are not really review, matrix $(g_{ij})$ likely not review
> Orientations might not be review

**HW 1 due on Feb 6.**
:::
#### Outline of topics

* Vector spaces
  * Linear independence, basis & dimension
  * Inner products and Cauchy-Schwarz inequality
  * Orthonormal bases
  * $g=(g_{ij})$ matrix
* Orientations
* Linear transformations (_might extend into Week 2_)

:::info
**Calculus (and some _analytic geometry_)**
- Week 2, [time=Feb 6 and 8, and part of Feb 13]

**Read:** Sections: 1-4, 1-5.
**HW 2 due on Feb 15.**
:::
#### Outline of topics
* Lines, planes, ~~spheres~~ (in $\mathbb R^3$)
* Parameterization
* Vector calculus
  * Chain rule in this setting
  * Product rules for inner products, and the cross product

## Unit 2: Regular curves (4-5 weeks)
> Mostly, curves will be in $\mathbb R^3$ ("space curves")

:page_with_curl: 4 homeworks (3 local theory, 1 global theory)

### Begin with _local theory_.

:::info
**Definitions & examples**
- Week 3, [time=Feb 13 and 15]

**Read:** Sections 2-1, 2-2. 
**HW 3 due on Feb 22.**
:::
#### Outline of topics

* Regular space curves, examples 
* Reparameterizations
* Arclength (_discussion of section, e.g. unit speed curves, extends to Week 4_)

:::info
**Arc length parameterized curves & Curvature**
- Weeks 4 and 5, [time=Feb 20, 22, and Feb 27]

**Read:** Sections 2-3. 
**HW 4 due on Mar 1.**
:::
:::info
**The Frenet-Serret apparatus**
- Second part of Week 5, [time=Mar 1]

**Read:** Section 2-3.
**HW 5 (on curvature, torsion and Frenet-Serret) due on Mar 8.**
:::
#### Outline of topics

* Using a.l.p. curves (unit speed curves)
* The curvature, $\kappa$, of a regular curve
* The Frenet-Serret apparatus $\{\kappa, \tau, \bf T, N, B\}$

:::info
**Frenet-Serret equations & Planes related to curves, Non-arclength formulae**
- Week 6, [time=Mar 6 and 8]

**Read:** Section 2-4, 2-6.
**HW 6 due on Mar. 13.**
:::
#### Outline of topics

* The Frenet-Serret equations (Theorem 4.2)
* The osculating plane and normal plane of a space curve
* Calculations with curves that are not a.l.p.

:::    success
**Test 1, Mar. 13 in class.** On Chap. 1 (not 1-2), Sections 2-1 through 2-4, and Section 2-6.
:::
### Some _global theory_.
:::info
**Line integrals, plane curvature, rotation index.**
- during Week 7, [time=Mar 15]

**Read:** Sections 3-1 and 3-2.
**HW 7 due on Mar. 27.**
:::
#### Outline of topics

* Green's Theorem
* Simple closed curves in the plane
* Plane curvature
* Hopf's Umlaufsatz and Rotation index

:::info
**The isoperimetric inequality**
> Might skip this topic
> 
- Week 8 (?), [time=Mar 27]
**Read:** Section 3-4.
:::

#### Outline of topics

* Proving the isoperimetric inequality

<!--
## Unit 3: Surfaces
:page_with_curl: 4 homeworks

:::info
**Open sets in $\mathbb R^2$**
- Week 8, [time=Oct 21]

**Read:** Section 4-1.
:::

:::info
**Coordinate patches, coordinate transformations**
- Week 9, [time=Oct 26 and 28]
**Read:** Section 4-1.
**HW 6: due on Nov. 2**
:::

#### Outline of topics

* Definition of a _coordinate patch_ (a _simple surface_) and examples
* _Coordinate transformations_ (analogue of reparameterizations, but for surfaces)
* The _tangent plane_ to simple surface at a point

:::info
**Regular surfaces & Metric coefficients and First Fundamental Form**
- Week 10, [time=Nov 2 and 4]
**Read:** Sections 4-2 and 4-3.
**HW 7: due Nov 11.**
:::

#### Outline of topics

* Definition of _regular surface_, some discussion of examples (incl. surfaces of revolution)
* The metric coefficients of a coordinate patch, computing arclength of curves on surface with metric coefficients.

:::info
**Normal & Geodesic Curvature**
- Week 11, [time=Nov 9 and 11]
**Read:** Section 4-4.
**HW 8: due on Nov 23.**
:::

#### Outline of topics

* Definition of _geodesic curvature_ and _normal curvature_. Some examples.
* Coefficients of the Second Fundamental Form.

:::info
**Christoffel Symbols, Gauss's formulas**
- Week 12 (and Tues before Thanksgiving), [time=Nov 16 and 23]
**Read:** More from Section 4-4.
**Exam 2 on Nov 18:** On Sections 2-6, 3-1,3-2,3-4, and 4-1 through 4-4.
:::

#### Outline of topics

* Christoffel symbols, Gauss' Formulas 
* The Christoffel symbols are intrinsic; geodesic curvature is intrinsic.

:::info
**Geodesics, the Second Fundamental form**
- Week 13, [time=Nov 30 and Dec 2]
**Read:** Section 4-5 and 4-7.
**HW 9: due on Dec 7**
:::

#### Outline of topics

* Geodesics of surfaces of revolution
* Distance minimizers are geodesics
* The second fundamental form & Weingarten map - introduction.

:::info
**Principal and Gaussian curvature & Gauss' Theorem Egregium**
- Week 14, [time=Dec 7 and Dec 9]
**Read:** Sections 4-8 and 4-9.
**HW 10 (short): due on Dec 9**
:::

#### Outline of topics

* Eigenvalues of the Weingarten map - Principal curvatures.
* Gaussian curvature
* Riemannian curvature tensor and Gauss' Theorem Egregium: Gaussian curvature is intrinsic.
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