Week 
Date 
Material 

Mon, Jan. 9 

Overview of differential geometry
 Euclid: ~300BC
 Descartes: ~1637
 Newton/Leibniz: ~1670
 (Euler, Lagrange, Monge, ...) 1700s
 Gauss: 1827: General Investigations on Curved Surfaces
 Elimination of infinitesimals (CauchyWeierstass ~18251861) and modern mathematics (set theory; CantorDedekindHilbertFregeRussell, ~1900)
 Riemann: 1854
 Einstein: 1915
 The modern era of differential/Riemannian geometry (1900s)

Brief overview of topics
 Parametrized curves/surfaces versus level curves/surfaces
 Tangent vectors/planes
 Curvature
 Intrinsic geometry and Theorema Egregrium
 Geodesics
 Global topology and GaussBonnet
 Many other little things along the way, and many examples
 Parametrized curves (Definition 1.1.1)
 Level curves and parametrization of (part of) a level curve (p .2)
 Parametrizing a parabola (Example 1.1.2)

Some alreadyknown concepts
 The set of real numbers and ntuples of real numbers
 Functions between sets
 Setbuilder notation
 Note: mere familiarity with and intuition about these basic concepts is enough; for the concepts introduced in this class, you must know the precise definitions.

1 
Wed, Jan. 11 

Review of two steps involved in finding a parametrization of a curve
 Definition of the image of a curve: im(γ)={γ(t)  t ∈ (α,β)}
 The two steps in proving that two steps are equal (⊂ and ⊃)
 Parametrizing the circle (Example 1.1.3)
 Parametrizing the astroid (Example 1.1.4)

Definition of smooth functions and smooth parametrized curves (p. 4)
 Smooth functions are closed under basic operations (addition, composition, etc.)
 From now we assume all parametrized curves are smooth

Tangent vector to a curve (Definition 1.1.5)
 Definition of tangent line

A curve with constant tangent vector is a straight line (Proposition 1.1.6)

Some alreadyknown concepts
 Derivatives and higher derivatives
 Limits
 Rules for derivatives (sum, product, quotient, chain rule)
 Trigonometric functions, their derivatives, and trig identities


Fri, Jan. 13 
 Proof of Proposition 1.1.6

The limaçon (Example 1.1.7)
 A curve can have two different tangent vectors at the same point at different times
 Thus it is an abuse of notation to write “the tangent vector at the point γ(t)”, but we do it anyway

Arclength (Definition 1.2.1)
 Arclengths starting at different points differ points differ by a constant
 The derivative of the arc length is the speed

Unitspeed curves (Definition 1.2.3)
 For a unit speed curve, the parameter is just the arclength, up to a constant
 The product rule for derivatives of dot products (p. 11)
 The tangent vector of a unitspeed curves is orthogonal to its derivative (Proposition 1.2.4)

Some alreadyknown concepts
 Integrals
 The fundamental theorem of calculus
 Other integration rules (substitution, integration by parts, etc.)
 Riemann sums
 Norm/length of a vector
 Dot product
 The norm squared of a vector is its dot product with itself


Mon, Jan. 16 
MLK’s birthday. No class! 
2 
Wed, Jan. 18 

Reparametrization of a curve (Definition 1.3.1)
 If γ_{1} is a reparametrization of γ_{2}, then γ_{2} is a reparametrization of γ_{1}.
 Warning: not every smooth bijection has a smooth inverse

Reparametrizing the circle (Example 1.3.2)
 Warning: given a parametrization γ of a level curve C, not every parametrization of C is a reparametrization of γ.
 Digression on “change of variables”
 Regular/singular point and regular curve (Definition 1.3.3)
 Any reparametrization of a regular curve is regular (Proposition 1.3.4)

Lemma: any smooth bijection with smooth inverse has a nonvanishing derivative
 and converesely, any smooth function with nonvanishing derivative is a bijection onto its image and has a smooth inverse


Fri, Jan. 20 

The concept of “parameter” (see this supplement)
 The derivative of a curve or function with respect to a parameter
 A parameter u is a unitspeed parameter if and only if the reparametrization with respect to u is a unit speed curve
 The arclength of a curve is a parameter if and only if the curve is regular (Propositions 1.3.61.3.7)
 Up to sign and a constant, the arclength is the only unit speed parameter (Corollary 1.3.7)

A unit speed parametrization can be difficult or impossible to compute
 Example 1.3.8: logarithmic spiral
 Example 1.3.9: the twisted cubic


Mon, Jan. 23 

Tperiodic curves and closed curves (Definition 1.4.1)
 Every curve is 0periodic
 A curve is Tperiodic if and only if it is (T)periodic, so we might as well always assume T≥0.
 The period of a closed curve (Definition 1.4.2)
 The length of a closed curve (p. 21)
 A unitspeed parametrization is closed, and the period is the length (p.21)
 Selfintersection (Definition 1.4.4)
 Example: the limaçon has one selfintersection (Example 1.4.5)
 Smooth multivariable functions (p. 23)
 Regular level curves (the defining function is smooth with nonvanishing gradient)
 For any point p on a regular level curve C, there is a regular parametrization of part of C passing through P (Theorem 1.5.1)

3 
Wed, Jan. 25 

Proof of Theorem 1.5.1 (except for the smoothness and regularity of the parametrized curve)
 If one introduces the notion of a “connected” curve, one can show that for a connected regular level curve, there is a regular parametrization of the whole curve
 For a regular parametrized curve, there is a piece of it near any point on it that is part of a regular level curve (Theorem 1.5.2)
 Curvature of unit speed curves (Definition 2.1.1)
 A circle of radius R has constant curvature 1/R (p. 31)


Fri, Jan. 27 

Curvature of regular curves (p. 31)
 Checked that it's independent of the unitspeed parameter used to define it
 Review of cross products
 Formula for curvature in terms of first and second derivative of γ (Proposition 2.1.2)
 Signed unit normal vector and signed curvature (p. 35)
 Review of rotation matrices


Mon, Jan. 30 
 Definition (2.2.2) and existence and uniqueness (Proposition 2.2.1) of the turning angle
 The turning angle is equal to the signed curvature (Proposition 2.2.3)

4 
Wed, Feb. 1 

Signed curvature of catenary (Example 2.2.4)
 Trick: the tangent tan(φ) of the turning angle is the quotient of the components of the tangent vector
 Total signed curvature of a closed curve (p. 39)
 The total signed curvature is a multiple of 2π (Corollary 2.2.5)
 Isometries and direct isometries of the plane (p. 39)
 Any prescribed signed curvature function determines a unitspeed curve, which is unique up to direct isometry (Theorem 2.2.6)


Fri, Feb. 3 

A plane curve with nonzero constant curvature is part of a circle (Example 2.2.7)
 In this case, signed curvature is plus or minus the curvature, since an integervalued continuous function on a connected interval must be constant
 A simple signed curvature function can lead to a complicated curve (Example 2.2.8)

If the curvature is not nonvanishing, it does not determine the curve up to isometry

In R^{3}, even if the curvature is nonvanishing, it does not determine the curve up to isometry
 Example: a helix and a circle both have constant curvature, but are obviously not related by an isometry (p. 46)
 Definition of principal normal vector, binormal vector, and of torsion (pp. 4647)

The unit tangent, principal normal, and binormal vector form an oriented/righthanded orthonormal basis at every point (p. 46)
 A basis is orthonormal if and only if the matrix with those vectors as columns or rows is orthogonal
 An ordered basis is oriented if and only if the associated matrix has determinant 1
 The product rule for cross products (p. 47)


Mon, Feb. 6 
 Formula for torsion in terms of the derivatives of γ (Proposition 2.3.1)
 (axb)·c is the determinant of the matrix with columns a,b,c
 Torsion of a helix (Example 2.3.2)
 Torsion vanishes if and only if the curve lies in a plane (Proposition 2.3.3)
 Review of equational form of a plane
 The FrenetSerret equations (Theorem 2.3.4)
 Skewsymmetric matrices (p. 51)

5 
Wed, Feb. 8 

The curvature and torsion determine a curve up to isometry (Theorem 2.3.6)
 Some nice ideas that we used in the proof, but that aren't essential knowledge for this class:
 A system of equations of the form X'(t)=AX(t) (with A and X matrices) can be solved using matrix exponentials
 If X'(t) is skewsymmetric for all t, then X(t) is orthogonal for all t (provided X(t_{0}) is orthogonal for some t_{0})


Fri, Feb. 10 

Simple closed curves (Definition 3.1.1)
 The limaçon is closed but not a simple closed curve (Example 3.1.3)

The Jordan Curve Theorem: the complement of a simple closed curve is the disjoint union of a bounded "interior" and an unbounded "exterior" (p. 55)
 Sketch of proof: first prove it for polygons by counting whether there are an even or odd number of points below a given point. And then approximate a general curve by a polygonal curve.
 For an ellipse, we can prove the theorem directly (Example 3.1.2: f)
 Definition of "positivelyoriented" based on the notion of "interior" coming from the Jordan Curve Theorem (p. 57)

Hopf's Umlaufsatz: the total signed curvature of a simple closed curve is ±2π, with the sign given by whether the curve is positively or negatively oriented. (Theorem 3.1.4)
 A proof along the lines sketched in class can be found here

Reminder on double integrals
 They are first defined on rectangles using Riemann sums
 They are extended to general bounded regions by multiplying with a characteristic function
 Fubini's theorem: they can be computed on rectangles, or on the area bounded by the graphs of two functions, using an interated integral
 Definition of the area bounded by a curve: the integral of the function 1 over its interior.
 Statement of the Isoperimetric Inequality (Theorem 3.2.2): A≤ℓ^{2}/4π


Mon, Feb. 13 
 Green's theorem (p. 58)

The definition of ∫f(x,y)dx + g(x,y)dy over a curve γ
 It is defined as ∫f(u(t),v(t))u'(t)dt + g(u(t),v(t))v'(t)dt over the period of γ, where u and v are the components of γ

Wirtinger's inequality (Proposition 3.2.3, but we used the version here)
 This involved some Fourier analysis: every smooth 2πperiodic function f(t) is a sum a_{0}/2 + Σ_{k≥1} a_{k}cos(kt) + b_{k}sin(kt)
 Also, the integral of f(t)^{2} is just the sum a_{0}^{2} + Σ_{k≥1} a_{k}^{2} + b_{k}^{2}
 Proof of isoperimetric inequality (Theorem 3.2.2, but we followed the proof here)

6 
Wed, Feb. 15 
 Intuitive notion of a surface: a subset of R^{3} that "looks like a piece of R^{2} in the vicinity of each point
 Open subsets of R^{n} (p. 68)
 Open and closed balls around points (p. 68)
 Continuous maps between subsets of R^{n} (with possibly different "n" for the domain and codomain)

Facts about continuous maps:
 A map is continuous if and only if each of its coordinate functions are continuous
 Continuous maps to R are closed under addition, subtraction, multiplication, division (if the denominator is nonvanishing), and constant maps are continuous
 The composite of two continuous maps is continuous
 Any map on X⊂R^{n} which is the restriction of a smooth (or even oncedifferentiable in each variable) map on R^{n} is continuous
 Summary, "if it looks continuous, it's continuous" and "almost every map you'll come across is continuous"
 Fact about open sets: any set defined by strict inequalities between continuous functions is open
 Homeomorphisms (p. 68)

Surfaces (Definition 4.1.1)
 Some auxiliary notions:
 Open subset of a set S⊂R^{n}
 Surface patches/parametrizations
 Atlases
 Every plane in R^{3} is a surface (Example 4.1.2)


Fri, Feb. 17 
 A cylinder is a surface (Example 4.1.3)
 A sphere is a surface (Example 4.1.4)

Warning: Pressley often write "parametrization" when he just mean "continous surjection onto a surface", rather than the official definition: "homeomorphism from an open subset of R^{2} to an open subset of the surface".
 In both the case of the cylinder and sphere, we found our atlas of surface patches by starting with a natural "parametrization" of the whole surface, and then restricting the domain so as to make it a homeomorphism.


Mon, Feb. 20 
President's day. No class! 
7 
Wed, Feb. 22 
 Some topology (see this supplement)

The circular cone is not a surface (Example 4.1.5)
 The reason is that any path from the "top half" of the cone to the "bottom half" must pass through the vertex, whereas in any open disk in the plane, any two points can be joined by a path missing the center; this shows that there cannot be a surface patch containing the center.
 However, if we remove the vertex, then it is a surface.
 Also, if we only take "one half" of the cone, it is also a surface (though not a smooth surface, a concept we will come to soon)
 Transition maps between surface patches (p. 74)


Fri, Feb. 24 
 Smoothness and partial derivatives of multivariable functions (p. 76)
 Regular/allowable surface patches (Definition 4.2.1)
 Smooth surfaces (Definition 4.2.2)
 The plane, cylinder and sphere are smooth (Examples 4.2.34.2.5)
 The transition maps of a smooth surface are smooth (Proposition 4.2.6); we will see the proof later

A diffeomorphism between open subsets of R^{n} is a smooth bijection with smooth inverse
 Given a diffeomorphism Φ:U→U' and a map σ:U→R^^{m}, the reparametrization of σ with respect to Φ is the map σ∘Φ^{1}:U'→R^^{m}
 Any reparametrization of a regular surface patch is regular (Proposition 4.2.7); we will see the proof next class
 The multivariable chain rule; next class, we will discuss this supplement


Mon, Feb. 27 
 Partial derivatives with respect to an arbitrary coordinate system (see this supplement)

Proof of Proposition 4.2.7 (any reparametrization of a regular surface patch is regular)
 Reminder: the determinant of an invertible matrix is nonzero
 We also used that (by the matrix version of the chain rule) the derivative of a diffeomorphism at each point is an invertible matrix
 There is a more direct proof of the proposition, using that if a matrix has linearly independent columns, so does any product of it with an invertible matrix
 Corollary of Proposition 4.2.6 (p. 79): any two surfaces patches, restricted to their overlapping regions, are reparametrizations of one another
 Convention from now on (p. 79): unless we say otherwise, we will assume all surfaces are smooth and connected

8 
Wed, Mar. 1 

Smooth maps between surfaces (p. 83)
 The book gives the case when each surface is covered by a single surface patch; for the general definition, we consider a surface patch on the first surface which is mapped into a given surface patch on the second surface.
 The notion of smoothness does not depend on which surface patch you use, because transition maps are smooth.
 Diffeomorphisms of surfaces (p. 83)
 Fact: smooth maps are continuous, hence diffeomorphisms are homeomorphisms.

Local diffeomorphisms (p. 83)
 A plane wrapping around a cylinder is a local diffeomorphism (Example 4.3.2)
 We define a map from a surface to R^{n} to be smooth if the map obtained by composing it with any regular surface patch is smooth. (Exercise 4.3.1)
 Tangent vectors to a surface at a point, and the tangent space to the surface at a point (Definition 4.4.1)


Fri, Mar. 3 
Midterm in class 

Mon, Mar. 6 

Every smooth curve on a surface is the composition of a smooth plane curve with a surface patch (p. 85)
 This follows from the Inverse Function Theorem

Inverse Function Theorem (Theorem 5.6.1)
 If a function F has an invertible derivative matrix DF_{p} at a point p, then F restricts to a diffeomorphism between an open subset containing p and an open subset containing F(p)

First corollary: if σ:U→V is a surface patch, then for each p∈V, the inverse σ^{1}:V→U extends to a smooth map on an open subset of p.
 That is, there is an open subset W⊂R^{3} containing p and a smooth map G:W→U with G(x)=σ^{1}(x) for x∈V.
 Second corollary: the transition functions between regular surface patches are smooth (Proposition 4.2.6)

The tangent space to a surface at a point is the span of the partial derivatives of any surface patch (Proposition 4.4.2)

9 
Wed, Mar. 8 

Any map between surfaces which is a restriction of a smooth map on an open subset of R^{3} is smooth
 More generally, it suffices for this to hold for the restriction of the function to an open neighbourhood of each point
 The proof uses the Inverse Function Theorem

Any smooth, regular, injective map σ:U→S from U⊂R^{3} open to a surface S is automatically a regular surface patch
 That is, it is not necessary to explicitly check the continuity of the inverse
 This is also proven using the Inverse Function Theorem
 Proof of Proposition 4.4.2 (stated last class)
 Corollary: the tangent space to a surface at any point is a 2dimensional linear subspace of R^{3} (i.e., a plane)
 The derivative D_{p}f of a map f between surfaces at a point p (Definition 4.4.3)
 Beginning of the proof that D_{p}f(v) doesn't depend on the chosen curve with tangent vector v

First lemma: the same description is valid for the derivative matrix D_{p}F of a smooth function F:U→V between open subsets U⊂R^{m} and V⊂R^{n}
 That is, if γ(0)=p and γ'(0)=v, then D_{p}F(v)=(Fᐤγ)'(0)
 The proof is immediate by the chain rule


Fri, Mar. 10 

Proof that D_{p}f(v) doesn't depend on the chosen curve with tangent vector v
 The proof used that the derivative D_{x}σ:R^{2}→T_{σ(x)}S of a regular chart σ is linear bijection
 The derivative of a map of surfaces at a point is a linear map (Proposition 4.4.4)
 The derivative of the identity is the identity, the derivative of the composite is the composite of the derivatives, and the derivative of a diffeormopshism invertible (Proposition 4.4.5)

A map whose derivative at every point is invertible is a local diffeomorphism (Proposition 4.4.5)


Mon, Mar. 13 
Spring recess. No class! 

Wed, Mar. 15 
Spring recess. No class! 

Fri, Mar. 17 
Spring recess. No class! 

Mon, Mar. 20 
 Normal vector to a surface at a point (p. 89)
 Standard unit normal of a surface patch (p. 89)
 Orientable surfaces (Definition 4.5.1)
 Oriented surface (p. 90)
 A surface is orientable if and only if it can be made into an oriented surface (Proposition 4.5.2)
 Convention: when considering an oriented surface, we will only consider those surface patches for which the standard unit normal agrees with the chosen orientation (p. 90)
 The Möbius strip is not orientable (Example 4.3.5)

10 
Wed, Mar. 22 

Review of some linear algebra facts
 Any linear subspace V⊂R^{3} has dimension 0, 1, 2, or 3
 If V⊂V'⊂R^{3} are both linear subspaces, then dim V≤dim V', with equality if and only V=V'
 The span of any nonzero vector is 1dimensional
 The span of any two independent vectors is 2dimensional
 (In particular, any two independent vectors in a tangent space to a surface span the whole tangent space.)
 If w is orthogonal to both v and w, then it is orthogonal to Span(v,w)
 If T:V→W is a linear map, then the image im(T)⊂W is a linear subspace
 If T:V→W is a linear map and dim(V)=dim(W) then T is injective iff it's surjective iff it's bijective

Any set S (locally) defined by a smooth function f whose gradient is nonvanishing on S is a smooth surface (Theorem 5.1.1)
 Also, the gradient f gives a normal vector to S at each point
 The sphere and circular cone minus the origin (Examples 5.1.25.1.3)

Quadric surfaces (Definition 5.2.1)
 Classification of quadric surfaces up to direct isometry (Theorem 5.2.2)


Fri, Mar. 24 
 Proof of Proposition 4.4.5, which was stated in class on Friday, March 10.
 Ruled surfaces (Example 5.3.1)
 Generalized cylinders (p. 105)
 Generalized cones (p. 106)
 Surfaces of revolution (Example 5.3.2)
 Compact subsets of R^{n} (p. 109)
 Planes, and open disks are not compact, but spheres and tori are (p. 110)

Informal definition of a compact surface of genus g
 Every compact surface is diffeomorphic to the compact surface of genus g for some g (Theorem 5.4.4)
 Corollary 5.4.5: every compact surface in R^{3} is orientable


Mon, Mar. 27 

"Substitution" interpretation of computing arclengths using first fundamental form
 To compute the integral over γ of ∫(Edu^{2}+2Fdudv+Gdv^{2})^{1/2}, simply express u and v as functions of t, find the corresponding expressions for du and dv, and "substitute" into the integral

First fundamental form of a surface of revolution (Example 6.1.3)
 Comptuation of the arc length of parallels and profile curves
 First fundamental form of a generalied cylinder (Example 6.1.4)
 First fundamental from of a generalized cone (Example 6.1.5)

11 
Wed, Mar. 29 
 Isometries (§6.2)
 Conformal mappings (§6.3)


Fri, Mae. 31 
 Equiareal mappings (§6.4)
 Curvature of surfaces (§7)
 The second fundamental form (§7.1)
 Gauss and Weingarten maps (§7.2)


Mon, Apr. 3 
 Normal and geodesic curvatures (§7.3)
 Parallel transport and covariant derivative (§7.4)

12 
Wed, Apr. 5 
 Gaussian and mean curvatures (§8.1)
 Principal curvatues (§8.2)


Fri, Apr. 7 
 Some surfaces of constant curvature (§8.3)
 Flat surfaces (§8.4)
 Gaussian curvature of compact surfaces (§8.6)


Mon, Apr. 10 
 Geodesics (§9.1)
 Geodesic equations (§9.2)
 Geodesics as shortest paths (§9.4)

13 
Wed, Apr. 12 
 Geodesic coordinates (§9.5)
 The Gauss and CodazziMainardi equations (§10.1)


Fri, Apr. 14 
 Theorema Egregium (§10.2)


Mon, Apr. 17 
 Surfaces of constant curvature (§10.3)

14 
Wed, Apr. 19 
 (Maybe: Geodesic mappings (§10.4))


Fri, Apr. 21 
 GaussBonnet for simple closed curves (§13.1)


Mon, Apr. 24 
 GaussBonnet for curvilinear polygons (§13.2)

15 
Wed, Apr. 26 
 Integration on compact surfaces (§13.3)


Fri, Apr. 28 
 GaussBonnet for compact surfaces (§13.4)
