| 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 (Cauchy-Weierstass ~1825-1861) and modern mathematics (set theory; Cantor-Dedekind-Hilbert-Frege-Russell, ~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 Gauss-Bonnet
- 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 already-known concepts
- The set of real numbers and n-tuples of real numbers
- Functions between sets
- Set-builder 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 already-known 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
-
Arc-length (Definition 1.2.1)
- Arc-lengths starting at different points differ points differ by a constant
- The derivative of the arc length is the speed
-
Unit-speed curves (Definition 1.2.3)
- For a unit speed curve, the parameter is just the arc-length, up to a constant
- The product rule for derivatives of dot products (p. 11)
- The tangent vector of a unit-speed curves is orthogonal to its derivative (Proposition 1.2.4)
-
Some already-known 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 γ1 is a reparametrization of γ2, then γ2 is a reparametrization of γ1.
- 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 non-vanishing derivative
- and converesely, any smooth function with non-vanishing 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 unit-speed parameter if and only if the reparametrization with respect to u is a unit speed curve
- The arc-length of a curve is a parameter if and only if the curve is regular (Propositions 1.3.6-1.3.7)
- Up to sign and a constant, the arc-length 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
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|
Mon, Jan. 23 |
-
T-periodic curves and closed curves (Definition 1.4.1)
- Every curve is 0-periodic
- A curve is T-periodic 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 unit-speed parametrization is closed, and the period is the length (p.21)
- Self-intersection (Definition 1.4.4)
- Example: the limaçon has one self-intersection (Example 1.4.5)
- Smooth multivariable functions (p. 23)
- Regular level curves (the defining function is smooth with non-vanishing 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 unit-speed 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 unit-speed curve, which is unique up to direct isometry (Theorem 2.2.6)
|
|
Fri, Feb. 3 |
-
A plane curve with non-zero constant curvature is part of a circle (Example 2.2.7)
- In this case, signed curvature is plus or minus the curvature, since an integer-valued 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 non-vanishing, it does not determine the curve up to isometry
-
In R3, even if the curvature is non-vanishing, 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. 46-47)
-
The unit tangent, principal normal, and binormal vector form an oriented/right-handed 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 Frenet-Serret equations (Theorem 2.3.4)
- Skew-symmetric 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 skew-symmetric for all t, then X(t) is orthogonal for all t (provided X(t0) is orthogonal for some t0)
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|
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 "positively-oriented" 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≤ℓ2/4π
|
|
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 2π-periodic function f(t) is a sum a0/2 + Σk≥1 akcos(kt) + bksin(kt)
- Also, the integral of f(t)2 is just the sum a02 + Σk≥1 ak2 + bk2
- 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 R3 that "looks like a piece of R2 in the vicinity of each point
- Open subsets of Rn (p. 68)
- Open and closed balls around points (p. 68)
- Continuous maps between subsets of Rn (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 non-vanishing), and constant maps are continuous
- The composite of two continuous maps is continuous
- Any map on X⊂Rn which is the restriction of a smooth (or even once-differentiable in each variable) map on Rn 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⊂Rn
- Surface patches/parametrizations
- Atlases
- Every plane in R3 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 R2 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.
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|
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)
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|
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 4.2.3-4.2.5)
- 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 Rn 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
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|
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 non-zero
- 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
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| 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 Rn 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)
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Fri, Mar. 3 |
Midterm in class |
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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 DFp 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⊂R3 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)
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| 9 |
Wed, Mar. 8 |
-
Any map between surfaces which is a restriction of a smooth map on an open subset of R3 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⊂R3 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 2-dimensional linear subspace of R3 (i.e., a plane)
- The derivative Dpf of a map f between surfaces at a point p (Definition 4.4.3)
- Beginning of the proof that Dpf(v) doesn't depend on the chosen curve with tangent vector v
-
First lemma: the same description is valid for the derivative matrix DpF of a smooth function F:U→V between open subsets U⊂Rm and V⊂Rn
- That is, if γ(0)=p and γ'(0)=v, then DpF(v)=(Fᐤγ)'(0)
- The proof is immediate by the chain rule
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|
Fri, Mar. 10 |
-
Proof that Dpf(v) doesn't depend on the chosen curve with tangent vector v
- The proof used that the derivative Dxσ:R2→Tσ(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)
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Mon, Mar. 13 |
Spring recess. No class! |
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Wed, Mar. 15 |
Spring recess. No class! |
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Fri, Mar. 17 |
Spring recess. No class! |
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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⊂R3 has dimension 0, 1, 2, or 3
- If V⊂V'⊂R3 are both linear subspaces, then dim V≤dim V', with equality if and only V=V'
- The span of any non-zero vector is 1-dimensional
- The span of any two independent vectors is 2-dimensional
- (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 non-vanishing 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 5.1.2-5.1.3)
-
Quadric surfaces (Definition 5.2.1)
- Classification of quadric surfaces up to direct isometry (Theorem 5.2.2)
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|
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 Rn (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 R3 is orientable
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Mon, Mar. 27 |
-
"Substitution" interpretation of computing arc-lengths using first fundamental form
- To compute the integral over γ of ∫(Edu2+2Fdudv+Gdv2)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)
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| 11 |
Wed, Mar. 29 |
- Isometries (§6.2)
- Conformal mappings (§6.3)
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Fri, Mae. 31 |
- Equiareal mappings (§6.4)
- Curvature of surfaces (§7)
- The second fundamental form (§7.1)
- Gauss and Weingarten maps (§7.2)
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Mon, Apr. 3 |
- Normal and geodesic curvatures (§7.3)
- Parallel transport and covariant derivative (§7.4)
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| 12 |
Wed, Apr. 5 |
- Gaussian and mean curvatures (§8.1)
- Principal curvatues (§8.2)
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Fri, Apr. 7 |
- Some surfaces of constant curvature (§8.3)
- Flat surfaces (§8.4)
- Gaussian curvature of compact surfaces (§8.6)
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Mon, Apr. 10 |
- Geodesics (§9.1)
- Geodesic equations (§9.2)
- Geodesics as shortest paths (§9.4)
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| 13 |
Wed, Apr. 12 |
- Geodesic coordinates (§9.5)
- The Gauss and Codazzi-Mainardi equations (§10.1)
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Fri, Apr. 14 |
- Theorema Egregium (§10.2)
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Mon, Apr. 17 |
- Surfaces of constant curvature (§10.3)
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| 14 |
Wed, Apr. 19 |
- (Maybe: Geodesic mappings (§10.4))
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Fri, Apr. 21 |
- Gauss-Bonnet for simple closed curves (§13.1)
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Mon, Apr. 24 |
- Gauss-Bonnet for curvilinear polygons (§13.2)
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| 15 |
Wed, Apr. 26 |
- Integration on compact surfaces (§13.3)
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Fri, Apr. 28 |
- Gauss-Bonnet for compact surfaces (§13.4)
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