Contents Strategy and Perspective

Supplemental Lecture 13

Differential Forms for Physics Students

III. Classical and Riemannian Differential Geometry

Abstract

Differential forms are employed to develop the theory of surfaces and manifolds. The method of moving frames of E. Cartan is introduced and the fundamental formulas of differential geometry are obtained in a coordinate-free fashion. This perspective leads to remarkably simple derivations of Gauss’ Theorem Egregium and the Gauss-Bonnet Theorem. The Gauss-Bonnet Theorem is developed as a structural equation of differential geometry without the need for coordinate systems or detailed formulas for the Gaussian curvature of surfaces or the geodesic curvature of curves. These ideas are generalized to higher dimensional manifolds. Topics in Riemannian and non-Euclidean geometry are introduced and studied.

This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.

Keywords: Differential Forms, Classical Differential Geometry, Riemannian Geometry, E.Cartan, Orthonormal Frames, Gaussian Curvature, Theorem Egregium, Gauss-Bonnet Theorem, Hypersurfaces, Manifolds, Non-Euclidean Geometry

Contents Strategy and Perspective...... 2

Moving Frames ...... 3

Surfaces in R3 ...... 6

Gaussian and Mean Curvatures ...... 7

Theorem Egregium ...... 9

Harmonic Functions ...... 11

Gauss-Bonnet Theorem: Global Version ...... 13

Gauss-Bonnet Theorem: Local Version ...... 14

Hypersurfaces in Rn+1 ...... 16

Intrinsic Geometry of Manifolds ...... 20

Non-Euclidean Geometry ...... 26

Appendix: Making Contact with Classical Differential and Riemannian Geometry ...... 28

References ...... 34

Strategy and Perspective. In our earlier discussions of classical differential geometry, we began by setting up a coordinate mesh on a smooth surface S embedded in three dimensional Euclidean space R3. We then used the coordinate mesh to calculate properties of the surface and argued that those properties did not actually depend on the particular mesh chosen but were properties of the surface itself, some intrinsic and others dependent on the surface’s embedding in R3. In several cases we found that this approach obscured the underlying geometry under a flood of coordinate-dependent derivatives and algebraic detail. The approach was computational oriented, rather than concept oriented.

By contrast, using differential forms we aim to manipulate geometric objects directly, find identities between them and only use coordinate systems to quantify those identities when necessary. A critical element which makes this approach successful and productive is the moving frames approach introduced by E. Cartan [1]. Specializing to frames traveling on surfaces gives us differential expressions for the geometry of the surface. The curvatures of curves and surfaces are formulated from this perspective. No coordinates are necessary and only geometrical relations result. The Gaussian and mean curvatures are introduced. The Theorem Egregium follows easily. Minimal surfaces are introduced. The Gauss-Bonnet Theorem is easily derived in both its global and local forms [2]. It is particularly interesting that the local form of the theorem, which took considerable detailed work to derive in classical differential geometry drops out here with little effort as a structural, fundamental result, without the need for detailed formulas for

either the Gaussian or geodesic curvatures. (This success illustrates one of the tenets of M. Spivak [3] which states that important theorems are those which are supported by formulations and definitions which make them almost trivial!) This single result illustrates the incisiveness of using differential forms and frames to study differential geometry. These results are generalized to hypersurfaces in higher dimensional Euclidean spaces. Finally, using the lessons learned from these exercises, the intrinsic geometry of manifolds is discussed without the need for an embedding space and the covariant derivative and Riemann curvature tensor are derived in the language of differential forms and many of their properties are derived. The non-Euclidean geometry of the Poincare Upper Half Plane model is illustrated from this point of view.

Moving Frames The use of moving frames in R3 was pioneered by E. Cartan. To begin, consider a point = ( ) 3 { } 3 , , in R and let there be an orthonormal frame at , , , [1]. As moves through𝑥𝑥⃗ R { } we𝑥𝑥 𝑦𝑦suppose𝑧𝑧 that the frame , , moves smoothly.𝑥𝑥 ⃗Consider𝑒𝑒⃗1 𝑒𝑒⃗2 𝑒𝑒an⃗3 infinitesimal𝑥𝑥⃗ change in : is a vector whose coefficients𝑒𝑒⃗1 𝑒𝑒⃗2 are𝑒𝑒⃗3 one-forms, as introduced in Supplementary Lecture 12,𝑥𝑥⃗

𝑑𝑑𝑥𝑥⃗ = + + (1)

1 1 2 2 3 3 We make the same observations for the𝑑𝑑𝑥𝑥⃗ triad,𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗

= + + (2)

𝑖𝑖 𝑖𝑖1 1 𝑖𝑖2 2 𝑖𝑖3 3 The one-forms are anti-symmetric𝑑𝑑 in𝑒𝑒⃗ the𝜔𝜔 indices𝑒𝑒⃗ i𝜔𝜔 and𝑒𝑒⃗ j. To𝜔𝜔 show𝑒𝑒⃗ this, differentiate =

, 𝜔𝜔𝑖𝑖𝑖𝑖 𝑒𝑒⃗𝑖𝑖 ∙ 𝑒𝑒⃗𝑗𝑗 𝑖𝑖 𝑗𝑗 𝛿𝛿 + = + = 0 (3)

𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 It is convenient to use matrix notation𝑑𝑑 to𝑒𝑒⃗ write∙ 𝑒𝑒⃗ these𝑒𝑒⃗ ∙ 𝑑𝑑 expressions𝑒𝑒⃗ 𝜔𝜔 𝜔𝜔more efficiently,

0 = , = ( , , ) , = 0 (4) 1 12 13 𝑒𝑒⃗ 𝜔𝜔 𝜔𝜔0 𝑒𝑒⃗ �𝑒𝑒⃗2� 𝜎𝜎 𝜎𝜎1 𝜎𝜎2 𝜎𝜎3 Ω �−𝜔𝜔12 𝜔𝜔13� 3 13 13 So, we have, 𝑒𝑒⃗ −𝜔𝜔 −𝜔𝜔

= , = , = (5) 𝑇𝑇 𝑑𝑑𝑥𝑥⃗ 𝜎𝜎𝑒𝑒⃗ 𝑑𝑑𝑒𝑒⃗ Ω𝑒𝑒⃗ Ω −Ω 3

where “T” indicates “transpose”. The anti-symmetric character of is expected: it is the generator of a rotation, an Orthogonal transformation, whose generatorsΩ have anti-symmetric representations. This point was discussed in the context of the Frenet-Serret equations in earlier lectures.

We are clearly not done. In order to integrate the differential equations, = , =

, we need to know how and vary with . These relations, call “Integrabilit𝑑𝑑𝑥𝑥y⃗ Conditions”,𝜎𝜎𝑒𝑒⃗ 𝑑𝑑𝑒𝑒⃗ ( ) Ωfollow𝑒𝑒⃗ from = 0 for the𝜎𝜎 exteriorΩ derivative𝑥𝑥⃗. First, from = 0, we learn, 2 𝑑𝑑 ( ) = ( ) = ( ) =𝑑𝑑 (𝑑𝑑𝑥𝑥⃗ ) = 0 (6)

𝑖𝑖 𝑖𝑖 Note that we will be using the 𝑑𝑑Einstein𝑑𝑑𝑥𝑥⃗ index𝑑𝑑 𝜎𝜎 𝑒𝑒 ⃗convention𝑑𝑑 𝜎𝜎𝑒𝑒⃗ in this𝑑𝑑𝑑𝑑 lecture,𝑒𝑒⃗ − 𝜎𝜎 ∧ 𝑑𝑑𝑒𝑒⃗ is short-hand for

. In addition, we have been careful with signs in Eq. 6: for the exterior𝜎𝜎𝑖𝑖𝑒𝑒⃗𝑖𝑖 derivative, ∑(𝑖𝑖 𝜎𝜎𝑖𝑖𝑒𝑒⃗𝑖𝑖 ) = + ( 1) . Since = , Eq. 6 can be written, deg 𝜆𝜆 𝑑𝑑 𝜆𝜆 ∧ 𝜇𝜇 𝑑𝑑𝑑𝑑 ∧ 𝜇𝜇 (− ) = (𝜆𝜆 ∧)𝑑𝑑𝑑𝑑 𝑑𝑑 𝑒𝑒⃗= (Ω𝑒𝑒⃗ ) = 0 (7) which implies, 𝑑𝑑 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑑𝑑 𝑒𝑒⃗ − 𝜎𝜎 ∧ Ω 𝑒𝑒⃗ 𝑑𝑑𝑑𝑑 − 𝜎𝜎 ∧ Ω 𝑒𝑒⃗

= (8)

Similarly, from ( ) = 0, we learn 𝑑𝑑𝑑𝑑 𝜎𝜎 ∧ Ω

𝑑𝑑 𝑑𝑑𝑒𝑒⃗ = (9)

The first two equalities in Eq. 5 are called𝑑𝑑Ω “structuralΩ ∧ Ω equations” and Eq.’s 8 and 9 are “Integrability Conditions”. A past example of integrability conditions were the Codazzi- Mainardi equations of classical differential geometry, discussed in Supplementary Lecture 9. They guaranteed the smoothness of surfaces in R3.

Let’s illustrate the formalism of moving frames with a familiar example: spherical coordinates, ( , , ). The position vector reads,

𝑟𝑟 𝜃𝜃 𝜑𝜑 = (sin cos , sin sin , cos )

And we can construct the moving𝑥𝑥⃗ orthonormal𝑟𝑟 𝜃𝜃 𝜑𝜑 frame,𝜃𝜃 𝜑𝜑 𝜃𝜃

= = (sin cos , sin sin , cos ) 𝜕𝜕𝑥𝑥⃗ 𝑒𝑒̂1 𝜃𝜃 𝜑𝜑 𝜃𝜃 𝜑𝜑 𝜃𝜃 𝜕𝜕𝜕𝜕 4

1 = = (cos cos , cos sin , sin ) 𝜕𝜕𝑥𝑥⃗ 𝑒𝑒̂2 𝜃𝜃 𝜑𝜑 𝜃𝜃 𝜑𝜑 − 𝜃𝜃 𝑟𝑟 𝜕𝜕𝜕𝜕 1 = = ( sin , cos , 0) sin 𝜕𝜕𝑥𝑥⃗ 𝑒𝑒̂3 − 𝜑𝜑 𝜑𝜑 Taking the differential of , 𝑟𝑟 𝜃𝜃 𝜕𝜕𝜕𝜕

𝑥𝑥⃗ = + + = ( ) + ( ) + ( sin ) 𝜕𝜕𝑥𝑥⃗ 𝜕𝜕𝑥𝑥⃗ 𝜕𝜕𝑥𝑥⃗ 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑒𝑒1̂ 𝑟𝑟𝑟𝑟𝑟𝑟 𝑒𝑒̂2 𝑟𝑟 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑒𝑒̂3 And we identify, 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

= = = sin

1 2 3 Next, we can differentiate the𝜎𝜎 moving𝑑𝑑𝑑𝑑 frame𝜎𝜎 and𝑟𝑟 𝑑𝑑 𝑑𝑑identify𝜎𝜎 the𝑟𝑟 skew𝜃𝜃 -𝑑𝑑symmetric𝑑𝑑 one-form ,

= , 𝜔𝜔𝑖𝑖𝑖𝑖 𝑑𝑑𝑒𝑒⃗𝑖𝑖 𝜔𝜔𝑖𝑖𝑖𝑖𝑒𝑒⃗𝑗𝑗 = + + 𝜕𝜕𝑒𝑒⃗1 𝜕𝜕𝑒𝑒⃗1 𝜕𝜕𝑒𝑒⃗1 𝑑𝑑𝑒𝑒⃗1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 0 + (cos cos , cos sin𝜕𝜕𝜕𝜕 , sin𝜕𝜕𝜕𝜕) + (𝜕𝜕𝜕𝜕sin sin , cos sin , 0)

1 𝑑𝑑𝑒𝑒⃗ 𝜃𝜃 𝜑𝜑 𝜃𝜃 = (𝜑𝜑 −) +𝜃𝜃(sin𝑑𝑑𝑑𝑑 −) 𝜑𝜑 𝜃𝜃 𝜑𝜑 𝜃𝜃 𝑑𝑑𝑑𝑑

1 2 3 And similarly, 𝑑𝑑𝑒𝑒⃗ 𝑑𝑑𝑑𝑑 𝑒𝑒⃗ 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑒𝑒⃗

= ( ) + (cos )

2 1 3 𝑑𝑑=𝑒𝑒⃗( sin−𝑑𝑑𝑑𝑑 𝑒𝑒)⃗ + ( 𝜃𝜃cos𝑑𝑑𝑑𝑑 𝑒𝑒⃗ )

3 1 2 And we identify from =𝑑𝑑𝑒𝑒⃗ , Eq.− 5, 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑒𝑒⃗ − 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑒𝑒⃗

𝑑𝑑𝑒𝑒⃗ Ω𝑒𝑒⃗ 0 sin = = 0 cos sin cos𝑑𝑑𝑑𝑑 𝜃𝜃0 𝑑𝑑𝑑𝑑 Ω �𝜔𝜔𝑖𝑖𝑖𝑖� � −𝑑𝑑𝑑𝑑 𝜃𝜃 𝑑𝑑𝑑𝑑� And finally, the familiar volume element− in𝜃𝜃 spherical𝑑𝑑𝑑𝑑 − coordinates𝜃𝜃 𝑑𝑑𝑑𝑑 is,

= sin 2 𝜎𝜎1 ∧ 𝜎𝜎2 ∧ 𝜎𝜎3 𝑟𝑟 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

Surfaces in R3 Now let’s use moving frames to describe surfaces S in R3 [1]. This is the set-up we choose: let the frame moves along the surface with always normal to the surface. Then

and span the tangent𝑒𝑒⃗ plane. As varies across the surface,𝑒𝑒⃗3 the orientation of over a unit𝑒𝑒⃗1 2 sphere𝑒𝑒⃗2 S varies, defining the Gauss𝑥𝑥⃗ map discussed in Supplementary Lecture 9. 𝑒𝑒The⃗3 Gauss map and its derivative, the Weingarten map, contain much of the geometry of the surface. Let’s see how the geometry of the surface is exposed using frames and forms. Note that we will be doing all of this without setting up a coordinate mesh!

Since is constrained to the surface S, lies in the tangent plane at each location, so

= 0 and, 𝑥𝑥⃗ 𝑑𝑑𝑥𝑥⃗ 3 𝜎𝜎 = + (10)

1 1 2 2 and is the surface area element𝑑𝑑 𝑥𝑥on⃗ S𝜎𝜎. 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗

1 2 𝜎𝜎 ∧Now𝜎𝜎 consider the equations for the changes in the moving frame as varies. We tailor 3 the notation to surfaces in R by writing, 𝑒𝑒⃗ 𝑥𝑥⃗

= + (11)

3 1 1 2 2 This notation mirrors Eq. 10 and brings𝑑𝑑𝑒𝑒⃗ out the𝜔𝜔 𝑒𝑒geometry⃗ 𝜔𝜔 𝑒𝑒⃗ in the Gauss map: provided by ( ) 2 and the derivative of the Gauss map, the Weingarten map, which is realized𝑆𝑆 → 𝑆𝑆 by the linear 2 ( ) 𝑒𝑒transformation⃗3 𝑥𝑥⃗ between the tangent plane of S at and the tangent plane of S at . 3 Eq. 11 is accompanied by, 𝑥𝑥⃗ 𝑒𝑒⃗ 𝑥𝑥⃗

= , = (12)

1 2 1 3 2 1 2 3 So, = with, 𝑑𝑑𝑒𝑒⃗ 𝜔𝜔�𝑒𝑒⃗ − 𝜔𝜔 𝑒𝑒⃗ 𝑑𝑑𝑒𝑒⃗ −𝜔𝜔�𝑒𝑒⃗ − 𝜔𝜔 𝑒𝑒⃗

𝑑𝑑𝑒𝑒⃗ Ω𝑒𝑒⃗ 0 = 0 (13) 1 𝜔𝜔� −0𝜔𝜔 Ω �− 𝜔𝜔� −𝜔𝜔2� 1 2 Now consider the integrability conditions,𝜔𝜔 Eq.𝜔𝜔 8 and 9, = and = . So,

𝑑𝑑𝑑𝑑 𝜎𝜎 ∧ Ω 𝑑𝑑Ω Ω ∧ Ω

0 = ( , , 0) = ( , , 0) 0 = 1 𝜔𝜔� −0𝜔𝜔 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑1 𝑑𝑑𝑑𝑑2 𝜎𝜎1 𝜎𝜎2 ∧ �− 𝜔𝜔� −𝜔𝜔2� ( ) , , + 𝜔𝜔 1 𝜔𝜔 2 2 1 1 1 2 2 Writing these results out,− 𝜎𝜎 ∧ 𝜔𝜔� 𝜎𝜎 ∧ 𝜔𝜔� 𝜔𝜔 ∧ 𝜎𝜎 𝜔𝜔 ∧ 𝜎𝜎

= , = , + = 0 (14)

1 2 2 1 1 1 2 2 Next, = becomes,𝑑𝑑𝑑𝑑 −𝜎𝜎 ∧ 𝜔𝜔� 𝑑𝑑𝑑𝑑 𝜎𝜎 ∧ 𝜔𝜔� 𝜔𝜔 ∧ 𝜎𝜎 𝜔𝜔 ∧ 𝜎𝜎

𝑑𝑑Ω Ω ∧ Ω0 0 0 0 = 0 0 1 1 1 𝑑𝑑𝜔𝜔� −𝑑𝑑0𝜔𝜔 𝜔𝜔� −0𝜔𝜔 𝜔𝜔� −0𝜔𝜔 �− 𝑑𝑑𝜔𝜔� −𝑑𝑑𝜔𝜔2� �− 𝜔𝜔� −𝜔𝜔2� ∧ �− 𝜔𝜔� −𝜔𝜔2� 1 2 1 2 1 2 which implies the𝑑𝑑𝑑𝑑 equations,𝑑𝑑𝜔𝜔 𝜔𝜔 𝜔𝜔 𝜔𝜔 𝜔𝜔

+ = 0 , = , = (15)

1 2 1 2 2 1 We will see that these 𝑑𝑑equations𝜔𝜔� 𝜔𝜔 ∧formulate𝜔𝜔 differential𝑑𝑑𝑑𝑑 𝜔𝜔� geometry∧ 𝜔𝜔 𝑑𝑑very𝑑𝑑 efficient−𝜔𝜔� ∧ ly.𝜔𝜔

Gaussian and Mean Curvatures Now let’s introduce and discuss the Gaussian and Mean curvatures on a surface S in the context of differential forms [1-2]. Gauss introduced the intrinsic curvature K by relating the surface element of S at , , to the same quantity on S2, . Since S is a two dimensional

space, there is only𝑥𝑥⃗ 𝜎𝜎 1one∧ 𝜎𝜎 independent2 two-form on it. Therefore,𝜔𝜔1 ∧ 𝜔𝜔2 and must be proportional at each . The proportionality factor, which is also 𝜎𝜎a1 function∧ 𝜎𝜎2 of𝜔𝜔 1,∧ is𝜔𝜔 the2 Gaussian curvature K, 𝑥𝑥⃗ 𝑥𝑥⃗

= (16)

1 2 1 2 This is the fundamental definition of K𝜔𝜔 that∧ 𝜔𝜔 was, in𝐾𝐾 𝜎𝜎fact,∧ 𝜎𝜎the starting point for Gauss himself. Recall the corresponding equation and discussion in classical differential geometry, Supplementary Lecture 9, Eq. 55 on page 39,

× = ×

𝑑𝑑𝑁𝑁��⃗𝑢𝑢 𝑑𝑑𝑁𝑁��⃗𝑣𝑣 𝐾𝐾𝐾𝐾𝑟𝑟⃗𝑢𝑢 𝑑𝑑𝑟𝑟⃗𝑣𝑣

This relation expresses the same idea as Eq. 16, it relates the surface area element on the surface to that on the unit sphere, the target space of the Gauss map ( ): , but it relies on a 3 2 coordinate mesh and vector cross products in R for its formulation.𝑁𝑁��⃗ 𝑥𝑥⃗ 𝑆𝑆 → 𝑆𝑆

Let’s comment on the dimensions of the quantities in Eq. 16. Each has the dimension

of length as we illustrated in the discussion of spherical coordinates and is 𝜎𝜎apparent𝑖𝑖 in Eq. 10. But the orthonormal frame is dimensionless, so = is also. Therefore, K has dimension

. We learned in classical𝑒𝑒⃗ differential geometry Ωthat 1�𝜔𝜔𝑖𝑖𝑗𝑗� sets the scale of lengths: for −2 𝐿𝐿distances small compare to 1 , the geometry is essentially⁄√𝐾𝐾 Euclidean, but for lengths comparable to and larger than⁄ √1𝐾𝐾 , the geometry is curved. We learned in General Relativity that its space-time manifold is essentially⁄√𝐾𝐾 Minkowskian for nearby events in space-time. This aspect of Riemannian manifolds is called the Equivalence Principle in the language of physicists: freely falling frames are locally inertial and the rules of special relativity apply.

Let’s return to the discussion of the Gaussian curvature K and the mean curvature H in R3. There are other two-forms on S. We saw that one candidate + is in fact zero.

However, is non-trivial so it too must be proportional𝜔𝜔1 ∧ 𝜎𝜎1 to𝜔𝜔 2 ∧ 𝜎𝜎2 . The proportionality𝜎𝜎1 ∧ 𝜔𝜔is2 defined− 𝜎𝜎2 ∧ to𝜔𝜔 1be twice the mean curvature H, which has the dimension𝜎𝜎1 ∧ 𝜎𝜎2 , −1 = 𝐿𝐿 (17)

1 2 2 1 1 2 To see that K and H are the familiar𝜎𝜎 ∧ 𝜔𝜔 curvatures− 𝜎𝜎 ∧ 𝜔𝜔 introduced𝐻𝐻 𝜎𝜎 ∧ in𝜎𝜎 classical differential geometry, let’s make the map between the tangent planes of S and S2 (Weingarten map) explicit. In particular, and can be written as linear superpositions of and because the set { } , is a complete𝜔𝜔1 𝜔𝜔 set2 of one forms on S. So, we can find coefficients,𝜎𝜎1 which𝜎𝜎2 depend on , so that,𝜎𝜎1 𝜎𝜎 2 𝑥𝑥⃗

= (18) 𝜔𝜔1 𝑝𝑝 𝑞𝑞 𝜎𝜎1 � 2� � � � 2� The linear transformation here is symmetric𝜔𝜔 because𝑞𝑞 𝑟𝑟 of 𝜎𝜎the constraint + = 0

which reads, 𝜔𝜔1 ∧ 𝜎𝜎1 𝜔𝜔2 ∧ 𝜎𝜎2

( + ) + ( + ) = ( + ) = 0 (19)

𝜎𝜎1 ∧ 𝑝𝑝𝜎𝜎1 𝑞𝑞𝜎𝜎2 𝜎𝜎2 ∧ 𝑞𝑞𝜎𝜎1 𝑟𝑟𝜎𝜎2 𝑞𝑞 𝜎𝜎1 ∧ 𝜎𝜎2 𝜎𝜎2 ∧ 𝜎𝜎1

The symmetric character of the linear transformation guarantees that its eigenvalues are real and its eigenvectors are perpendicular. Inserting the transformation Eq. 18 into Eq. 16 and 17, the definitions of K and H, we find,

= ( + ) = (20) 1 2 𝐻𝐻 2 𝑝𝑝 𝑟𝑟 𝐾𝐾 𝑝𝑝𝑝𝑝 − 𝑞𝑞 So, H is half the trace of the transformation and K is its determinant. Both of these quantities are independent of the basis leading to this representation. In fact, we can diagonalize the transformation and label the eigenvalues and . Then,

1 2 = ( +𝜅𝜅 ) 𝜅𝜅 = (21) 1 𝐻𝐻 2 𝜅𝜅1 𝜅𝜅2 𝐾𝐾 𝜅𝜅1𝜅𝜅2 These relations were derived in the textbook and various Supplementary Lectures. They are essential in differential geometry.

Theorem Egregium We now have the ingredients to show that K is an intrinsic property of the surface S and does not depend on how it is embedded in R3. In other words, K is determined just by measuring lengths and angles on the surface S itself. The key relations are + = 0 and =

, which can be combined, 𝑑𝑑𝜔𝜔� 𝜔𝜔1 ∧ 𝜔𝜔2 𝜔𝜔1 ∧ 𝜔𝜔2 1 2 𝐾𝐾 𝜎𝜎 ∧ 𝜎𝜎 + = 0 (22)

1 2 So, once we know , and , we can𝑑𝑑𝜔𝜔� calculate𝐾𝐾 𝜎𝜎 ∧ K.𝜎𝜎 But we know that,

1 2 𝜔𝜔� 𝜎𝜎 𝜎𝜎 = , =

1 2 2 1 from Eq. 14, so can be found given𝑑𝑑𝑑𝑑 − and𝜎𝜎 ∧ 𝜔𝜔�. In𝑑𝑑 particular𝑑𝑑 𝜎𝜎 ∧ 𝜔𝜔� can be written as a linear

superposition of𝜔𝜔 � and , so 𝜎𝜎1 𝜎𝜎2 𝜔𝜔� 1 2 𝜎𝜎 𝜎𝜎 = + (23)

1 2 implying, 𝜔𝜔� 𝑎𝑎𝜎𝜎 𝑏𝑏𝜎𝜎

= = (24)

1 1 2 2 1 2 which can be solved for a and 𝑑𝑑b𝑑𝑑, thereby𝑎𝑎𝜎𝜎 producing∧ 𝜎𝜎 𝑑𝑑𝑑𝑑 . 𝑏𝑏𝜎𝜎 ∧ 𝜎𝜎

𝜔𝜔� 9

This establishes that K is an intrinsic property of the surface. Explicit formulas for K can be derived by placing a coordinate mesh on the surface. Let’s carry out this exercise and derive a famous formula for the curvature K with much(!) less effort than it took us in the context of classical differential geometry. Recall that if we use an orthogonal mesh on S so that its metric reads = + , then we derived, 2 2 2 𝑑𝑑𝑑𝑑 𝐸𝐸𝑑𝑑𝑑𝑑 𝐺𝐺𝑑𝑑𝑑𝑑 = + (25) 1 𝜕𝜕 1 𝜕𝜕√𝐸𝐸 𝜕𝜕 1 𝜕𝜕√𝐺𝐺 𝐾𝐾 − √𝐸𝐸𝐸𝐸 �𝜕𝜕𝜕𝜕 �√𝐺𝐺 𝜕𝜕𝜕𝜕 � 𝜕𝜕𝜕𝜕 �√𝐸𝐸 𝜕𝜕𝜕𝜕 �� Now let’s use differential forms to obtain the same result in three steps [2]! The surface is described by a position function ( , ). We make the orthonormal frame = and

= with = and 𝑥𝑥⃗ 𝑢𝑢= 𝑣𝑣 . (In all these formulas the subscript𝑒𝑒⃗1 𝑥𝑥 ⃗u𝑢𝑢 ⁄or√ v𝐸𝐸 means 𝑒𝑒differentiation⃗2 𝑥𝑥⃗𝑣𝑣⁄√𝐺𝐺 with𝐸𝐸 respect𝑥𝑥⃗𝑢𝑢 ∙ 𝑥𝑥to⃗𝑢𝑢 that coordinate𝐺𝐺 𝑥𝑥⃗𝑣𝑣 ∙ 𝑥𝑥⃗ 𝑣𝑣variable: standard vector calculus notation.). Now we can calculate the associated co-frame, ( , ), using the definition, = + =

+ , 𝜎𝜎1 𝜎𝜎2 𝑑𝑑𝑥𝑥⃗ 𝑥𝑥⃗𝑢𝑢𝑑𝑑𝑑𝑑 𝑥𝑥⃗𝑣𝑣𝑑𝑑𝑑𝑑 1 1 2 2 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ = = (26)

1 2 Next we calculate from the 𝜎𝜎relation√𝐸𝐸s 𝑑𝑑𝑑𝑑 = 𝜎𝜎 √𝐺𝐺 ,𝑑𝑑𝑑𝑑 = ,

1 2 2 1 𝜔𝜔� = 𝑑𝑑𝑑𝑑 =−𝜎𝜎 ∧ 𝜔𝜔� 𝑑𝑑=𝑑𝑑 𝜎𝜎 ∧ 𝜔𝜔�

𝑑𝑑𝑑𝑑1 �√𝐸𝐸�𝑣𝑣𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 −𝜎𝜎2 ∧ 𝜔𝜔� −√𝐺𝐺 𝑑𝑑𝑑𝑑 ∧ 𝜔𝜔� = = =

𝑑𝑑𝑑𝑑2 � 𝐺𝐺�𝑢𝑢𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝜎𝜎1 ∧ 𝜔𝜔� 𝐸𝐸 𝑑𝑑𝑑𝑑 ∧ 𝜔𝜔� which implies, √ √

= + (27) �√𝐸𝐸�𝑣𝑣 �√𝐺𝐺�𝑢𝑢 𝜔𝜔� − √𝐺𝐺 𝑑𝑑𝑑𝑑 √𝐸𝐸 𝑑𝑑𝑑𝑑 But we have + = 0, and substituting Eq. 26 and 27 into this relation,

𝑑𝑑𝜔𝜔� 𝐾𝐾 𝜎𝜎1 ∧ 𝜎𝜎2 = + 𝜕𝜕 �√𝐸𝐸�𝑣𝑣 𝜕𝜕 �√𝐺𝐺�𝑢𝑢 𝑑𝑑𝜔𝜔� − � � 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 � � 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 √𝐺𝐺 𝜕𝜕𝜕𝜕 √𝐸𝐸 = + = = 𝜕𝜕 �√𝐸𝐸�𝑣𝑣 𝜕𝜕 �√𝐺𝐺�𝑢𝑢 𝑑𝑑𝜔𝜔� �𝜕𝜕𝜕𝜕 � √𝐺𝐺 � 𝜕𝜕𝜕𝜕 � √𝐸𝐸 �� 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 −𝐾𝐾 𝜎𝜎1 ∧ 𝜎𝜎2 −√𝐸𝐸𝐺𝐺 𝐾𝐾𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 and Eq. 25 falls out! Done.

Eq. 25 was central to many of the illustrations in Supplementary Lecture 9. The simple derivation of Eq. 25 using forms shows that manipulating these geometric objects has real advantages. We will discuss an even better(!) example when we come to the Gauss-Bonnet Theorem.

Let’s end this section with a few more calculations that will be useful in applications [1]. Consider the vector cross product × for in the tangent plane of a surface. Since =

+ consists of one-forms𝑑𝑑 and𝑥𝑥⃗ vectors,𝑑𝑑𝑥𝑥⃗ 𝑑𝑑we𝑥𝑥⃗ must be careful with the order of operations𝑑𝑑𝑥𝑥⃗ 𝜎𝜎when1𝑒𝑒⃗1 doing𝜎𝜎2𝑒𝑒⃗2 calculations,

× = ( + ) × ( + ) = ( × ) + ( × )

1 1 2 2 1 1 2 2 1 2 1 2 2 1 2 1 𝑑𝑑𝑥𝑥⃗ × 𝑑𝑑𝑥𝑥⃗ = 2𝜎𝜎( 𝑒𝑒⃗ 𝜎𝜎) 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒 ⃗ 𝑒𝑒 ⃗ 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒⃗ 𝑒𝑒 ⃗ (28)

1 2 3 where we𝑑𝑑𝑥𝑥⃗ note𝑑𝑑 𝑥𝑥the⃗ appearance𝜎𝜎 ∧ 𝜎𝜎 of𝑒𝑒⃗ the vectorial surface area. We can write this expression in terms of coordinates,

× = ( , , ) × ( , , ) = 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑� × = 2( , , ) = 2(𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑) 𝑑𝑑 𝑑𝑑 (29)

1 2 3 Similarly, 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒⃗

× = 2 ( )

3 1 2 3 And, 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑒𝑒⃗ 𝐻𝐻 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒⃗

× = 2 ( )

3 3 1 2 3 𝑑𝑑𝑒𝑒⃗ 𝑑𝑑𝑒𝑒⃗ 𝐾𝐾 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒⃗

These expressions and ones like them will be useful in our next discussion on harmonic functions.

Harmonic Functions Let’s do a simple application of these ideas [1]. We want to find the shape of minimal surfaces. These are surfaces which have vanishing mean curvatures. Soap bubbles are examples of

minimal surfaces. Soap bubbles can be supported by loops, as every child knows, and can also be created without a boundary.

First let’s recall the expression of the Laplacian, = + + , in 3 2 2 2 2 2 2 2 the language of differential forms. Given a function in R∇, we 𝜕𝜕have⁄𝜕𝜕 𝜕𝜕its differential,𝜕𝜕 ⁄𝜕𝜕𝜕𝜕 𝜕𝜕 ⁄𝜕𝜕𝜕𝜕 = + +

𝑥𝑥 𝑦𝑦 𝑧𝑧 where = , etc. Take the Hodge𝑑𝑑𝑑𝑑 star𝑓𝑓 𝑑𝑑 𝑑𝑑of this𝑓𝑓 𝑑𝑑 one𝑑𝑑 fo𝑓𝑓rm𝑑𝑑𝑑𝑑 in R3,

𝑥𝑥 𝑓𝑓 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = + +

𝑥𝑥 𝑦𝑦 𝑧𝑧 And finally, applying the exterior∗ 𝑑𝑑𝑑𝑑 derivative𝑓𝑓 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 to this𝑓𝑓 𝑑𝑑 𝑑𝑑result,∧ 𝑑𝑑𝑑𝑑 𝑓𝑓 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑

= + +

𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 𝑑𝑑 ∗ 𝑑𝑑𝑑𝑑 𝑓𝑓 𝑑𝑑=𝑑𝑑 ∧( 𝑑𝑑𝑑𝑑+∧ 𝑑𝑑𝑑𝑑 +𝑓𝑓 )𝑑𝑑 𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 =𝑓𝑓( 𝑑𝑑𝑑𝑑 )∧ 𝑑𝑑 𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 2 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 Let’s apply this𝑑𝑑 result∗ 𝑑𝑑𝑑𝑑 to functions𝑓𝑓 𝑓𝑓 defined𝑓𝑓 𝑑𝑑on𝑑𝑑 ∧the𝑑𝑑 𝑑𝑑surface∧ 𝑑𝑑𝑑𝑑 S. Then∇ 𝑓𝑓 its𝜎𝜎 differential is, using a coordinate mesh { , },

𝑢𝑢 𝑣𝑣 = +

𝑢𝑢 𝑣𝑣 Then applying the Hodge operator in R2𝑑𝑑𝑑𝑑, 𝑓𝑓 𝑑𝑑𝑢𝑢 𝑓𝑓 𝑑𝑑𝑑𝑑

𝑢𝑢 𝑣𝑣 (Be careful with signs: In R2, = ∗ 𝑑𝑑𝑑𝑑 and,𝑓𝑓 𝑑𝑑𝑑𝑑 −=𝑓𝑓 𝑑𝑑𝑑𝑑. Consult the previous Supplementary

Lecture 12 for an introduction ∗to𝑑𝑑𝑑𝑑 the Hodge−𝑑𝑑𝑑𝑑 duality∗ 𝑑𝑑𝑑𝑑 * operator.).𝑑𝑑𝑑𝑑 And finally, apply the exterior derivative,

= ( ) = ( + ) = ( ) 2 𝑢𝑢 𝑣𝑣 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 The same𝑑𝑑 ∗ ideas𝑑𝑑𝑑𝑑 apply𝑑𝑑 𝑓𝑓 𝑑𝑑𝑑𝑑to vectors− 𝑓𝑓 𝑑𝑑𝑑𝑑 on S, 𝑓𝑓 = 𝑓𝑓 𝑑𝑑𝑑𝑑+ ∧ 𝑑𝑑𝑑𝑑, so ∇ 𝑓𝑓 =𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 . Notice

that, 𝑑𝑑𝑥𝑥⃗ 𝜎𝜎1𝑒𝑒⃗1 𝜎𝜎2𝑒𝑒⃗2 ∗ 𝑑𝑑𝑥𝑥⃗ 𝜎𝜎2𝑒𝑒⃗1 − 𝜎𝜎1𝑒𝑒⃗2

× = ( + ) × =

3 1 1 2 2 3 2 1 1 2 So, we have the identity, 𝑑𝑑𝑥𝑥⃗ 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ − 𝜎𝜎 𝑒𝑒⃗

= ×

∗ 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑥𝑥⃗ 𝑒𝑒⃗3 12

And, using the remark after Eq. 29 above, × = 2 ( ) ,

3 1 2 3 = 𝑑𝑑×𝑥𝑥⃗ 𝑑𝑑=𝑒𝑒⃗ 2 𝐻𝐻( 𝜎𝜎 ∧ 𝜎𝜎) 𝑒𝑒⃗

3 1 2 3 Collecting, 𝑑𝑑 ∗ 𝑑𝑑𝑥𝑥⃗ −𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝑒𝑒⃗ − 𝐻𝐻 𝜎𝜎 ∧ 𝜎𝜎 𝑒𝑒⃗

= ( , , ) = 2 2 2 2 2 3 This is the result we wanted: A minimal∇ 𝑥𝑥⃗ ∇surface𝑥𝑥 ∇ 𝑦𝑦 ( ∇ =𝑧𝑧 0, locally)− 𝐻𝐻𝑒𝑒⃗ is a surface whose coordinate

fuctions are harmonic. So, each point on the surface𝐻𝐻 is a saddle point. There is much more to be said about minimal surfaces from the perspectives of geometry and the calculus of variations (minimax principles). The reader should consult the references in [2] to get started.

Gauss-Bonnet Theorem: Global Version Let’s return to the Gauss map, , taking S to S2. The Gaussian curvature K is defined as the

( 3 ) 2 ( ) ratio of the area elements on S 𝑥𝑥⃗ → 𝑒𝑒⃗ and S [1-2], 1 2 1 2 𝜎𝜎 ∧ 𝜎𝜎 = 𝜔𝜔 ∧ 𝜔𝜔 (16)

1 2 1 2 As varies over S, varies over S2. 𝜔𝜔By ∧elementary𝜔𝜔 𝐾𝐾 𝜎𝜎 theore∧ 𝜎𝜎 ms in topology, we know that if 2 varies𝑥𝑥⃗ over S, then 𝑒𝑒⃗3 varies over S an integral number of times, called the degree of the map,𝑥𝑥⃗ n. So, integrating Eq. 𝑒𝑒16,⃗3

= 4 (30)

𝑆𝑆 1 2 which is a form of the Global Gauss-Bonnet∫ 𝐾𝐾 𝜎𝜎 Theorem∧ 𝜎𝜎 𝜋𝜋derived𝑛𝑛 in Supplementary Lecture 3 and 11 in the context of traditional classical differential geometry. In particular, if S is a closed, convex surface, then varies over S2 exactly once, so then

3 𝑒𝑒⃗ = 4

𝑆𝑆 1 2 The generalization to a smooth surface∫ with𝐾𝐾 𝜎𝜎 g ∧holes𝜎𝜎 (genus)𝜋𝜋 is,

= 4 (1 ) (31)

∫𝑆𝑆 𝐾𝐾 𝜎𝜎1 ∧ 𝜎𝜎2 𝜋𝜋 − 𝑔𝑔

In particular, for a torus, = 0 which we showed explicitly in Supplementary Lecture

11 where K was calculated∫𝑆𝑆 from𝐾𝐾𝜎𝜎1 ∧the𝜎𝜎 2principle curvatures on the surface, = , and the integral was done by elementary means. 𝐾𝐾 𝜅𝜅1𝜅𝜅2 The power of Eq. 31, the global form of the Gauss-Bonnet Theorem, is that it holds even as S is distorted and K changes smoothly. The theorem expresses a global topological invariant which is the Euler characteristic of the surface. See previous Supplementary Lectures for more!

Gauss-Bonnet Theorem: Local Version There is another very useful form of the Gauss-Bonnet Theorem, pioneered the French mathematician by Pierre Ossian Bonnet, which applies to regions on a surface that are bounded by a curve C. If the curve is smooth, continuous and differentiable, then it reads [2],

+ = 2 (32)

𝑀𝑀 𝜕𝜕𝜕𝜕 𝑔𝑔 where M labels a connected region on∫ the𝐾𝐾 surface,𝑑𝑑𝑑𝑑 ∫ 𝜅𝜅 is𝑑𝑑𝑑𝑑 its boundary,𝜋𝜋 the curve C, and is its

geodesic curvature. We used this form of the Gauss𝜕𝜕𝜕𝜕-Bonnet Theorem and its generalization𝜅𝜅𝑔𝑔 to piece-wise smooth curves (curves with a finite number of “kinks”, abrupt changes in direction: think of a triangle) to discuss the non-Euclidean geometry on curved surfaces. The traditional derivations of Eq. 32 are very detailed and require the formulas for K and on a coordinate

mesh describing the surface. By contrast, differential forms provide a different𝜅𝜅𝑔𝑔 perspective of the theorem and allow a short, incisive derivation which does not require such detail. The crucial element in the proof is simply the definition of K and the integrability condition,

+ = + = 0 (33)

1 2 1 2 To begin, let’s describe𝑑𝑑 𝜔𝜔the� curve𝜔𝜔 ∧ C𝜔𝜔 that 𝑑𝑑bounds𝜔𝜔� 𝐾𝐾 the𝜎𝜎 ∧region𝜎𝜎 M. Along the curve there is a unit tangent ( ), parametrized by arc-length s, and there is a second unit vector which lies in

the tangent plane𝑡𝑡⃗ 𝑠𝑠 to the surface at that location and is perpendicular to ( ). The 𝑘𝑘rate�⃗ of change of ( ), , has a component in the tangent plane which points in the𝑡𝑡⃗ 𝑠𝑠 direction and has a magnitude𝑡𝑡⃗ 𝑠𝑠 𝑑𝑑𝑡𝑡⃗ ⁄which𝑑𝑑𝑑𝑑 defines the geodesic curvature . We can write the rate of change𝑘𝑘�⃗ of both ( ) and ( ), projected onto the tangent plane, 𝜅𝜅𝑔𝑔 𝑡𝑡⃗ 𝑠𝑠

𝑘𝑘�⃗ 𝑠𝑠

0 = 0 (34) 𝐷𝐷 𝑔𝑔 𝑡𝑡⃗ 𝜅𝜅 𝑡𝑡⃗ 𝑑𝑑𝑑𝑑 � � � 𝑔𝑔 � � � where we use the notation to indicate𝑘𝑘�⃗ the projection−𝜅𝜅 of𝑘𝑘 �⃗ onto the tangent plane. (This is the covariant derivative, 𝐷𝐷as⁄ introduced𝑑𝑑𝑑𝑑 in the lectures on classical𝑑𝑑⁄𝑑𝑑𝑑𝑑 differential geometry and Riemannian manifolds.) The 2 × 2 matrix in Eq. 34 is anti-symmetric to guarantee that the pair ( ), ( ) remain orthonormal as s varies. We also can relate ( ), ( ) to the pair of

�orthonormal𝑡𝑡⃗ 𝑠𝑠 𝑘𝑘�⃗ 𝑠𝑠 � vectors that define the coordinate mesh, { , }: they�𝑡𝑡⃗ are𝑠𝑠 𝑘𝑘�related⃗ 𝑠𝑠 � by a rotation, 1 2 cos sin𝑒𝑒⃗ 𝑒𝑒⃗ = sin cos 1 𝑒𝑒⃗ 𝜃𝜃 − 𝜃𝜃 𝑡𝑡⃗ � 2� � � � � So, the rate of change of { , } along𝑒𝑒⃗ C, projected𝜃𝜃 onto𝜃𝜃 the tangent𝑘𝑘�⃗ plane is,

𝑒𝑒⃗1 𝑒𝑒⃗2 = sin cos + cos sin 𝐷𝐷𝑒𝑒⃗1 𝑑𝑑𝑑𝑑 𝐷𝐷𝑡𝑡⃗ 𝐷𝐷𝑘𝑘�⃗ 𝑑𝑑𝑑𝑑 �−𝑡𝑡⃗ 𝜃𝜃 − 𝑘𝑘�⃗ 𝜃𝜃� 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜃𝜃 − 𝑑𝑑𝑑𝑑 𝜃𝜃 = + cos sin 𝐷𝐷𝑒𝑒⃗1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 − 𝑒𝑒⃗2 𝑑𝑑𝑑𝑑 �𝑘𝑘�⃗ 𝜅𝜅𝑔𝑔� 𝜃𝜃 − �−𝑡𝑡⃗ 𝜅𝜅𝑔𝑔� 𝜃𝜃 = + sin + cos = + = 𝐷𝐷𝑒𝑒⃗1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 − 𝑒𝑒⃗2 𝑑𝑑𝑑𝑑 �𝑡𝑡⃗ 𝜃𝜃 𝑘𝑘�⃗ 𝜃𝜃�𝜅𝜅𝑔𝑔 − 𝑒𝑒⃗2 𝑑𝑑𝑠𝑠 𝑒𝑒⃗2𝜅𝜅𝑔𝑔 𝑒𝑒⃗2 �𝜅𝜅𝑔𝑔 − 𝑑𝑑𝑑𝑑� But we had an expression, Eq. 12, for the rate of change of in terms of the one-forms on the surface, 𝑒𝑒⃗1

1 2 1 3 So, we identify, 𝑑𝑑𝑒𝑒⃗ 𝜔𝜔�𝑒𝑒⃗ − 𝜔𝜔 𝑒𝑒⃗

= (35) 𝑑𝑑𝑑𝑑 𝜔𝜔� �𝜅𝜅𝑔𝑔 − 𝑑𝑑𝑑𝑑� 𝑑𝑑𝑑𝑑 The Gauss-Bonnet Theorem Eq. 32 now follows from Stoke’s Theorem,

� 𝑑𝑑𝜔𝜔� � 𝜔𝜔� 𝑀𝑀 𝜕𝜕𝜕𝜕 Using Eq. 33,

− � 𝐾𝐾𝜎𝜎1 ∧ 𝜎𝜎2 ��𝜅𝜅𝑔𝑔𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑑𝑑� 𝑀𝑀 𝜕𝜕𝜕𝜕 15

which we collect,

+ = = 2

� 𝐾𝐾 𝑑𝑑𝑑𝑑 � 𝜅𝜅𝑔𝑔𝑑𝑑𝑑𝑑 � 𝑑𝑑𝑑𝑑 𝜋𝜋 𝑀𝑀 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 and we are done! The simplicity of this derivation indicates that the Gauss-Bonnet Theorem is a structural statement about surfaces and is fundamental. It is discussed at length in other lectures in this series.

Hypersurfaces in Rn+1 Now let’s turn to hypersurfaces in Rn+1 before we discuss Riemannian geometry [1]. A hypersurface means a n-dimensional manifold M. At the point on M, the orthonormal set { } , , … , spans its tangent space. If is the unit normal to𝑥𝑥 ⃗the tangent plane at , then { } n+1 𝑒𝑒⃗1, 𝑒𝑒⃗2, … , 𝑒𝑒⃗𝑛𝑛, make up an orthonormal𝑛𝑛� basis of R . Since lies in the tangent 𝑥𝑥plane,⃗ one can𝑒𝑒⃗1 write,𝑒𝑒⃗2 𝑒𝑒⃗𝑛𝑛 𝑛𝑛�⃗ 𝑑𝑑𝑥𝑥⃗

= + + + (36)

1 1 2 2 𝑛𝑛 𝑛𝑛 where are one-forms on M. 𝑑𝑑 𝑥𝑥⃗ 𝜎𝜎 𝑒𝑒⃗ 𝜎𝜎 𝑒𝑒⃗ ⋯ 𝜎𝜎 𝑒𝑒⃗

𝑖𝑖 𝜎𝜎 Now let’s turn to the variation of members of the basis, { , , … , , }. Since,

1 2 𝑛𝑛 = = 0 = 1𝑒𝑒⃗ 𝑒𝑒⃗ 𝑒𝑒⃗ 𝑛𝑛�⃗

𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖 we have the differentials,𝑒𝑒⃗ ∙ 𝑒𝑒⃗ 𝛿𝛿 𝑒𝑒⃗ ∙ 𝑛𝑛�⃗ 𝑛𝑛�⃗ ∙ 𝑛𝑛�⃗

+ = 0 + = 0 = 0

𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑖𝑖 So, if we write,𝑑𝑑 𝑒𝑒⃗ ∙ 𝑒𝑒⃗ 𝑒𝑒⃗ ∙ 𝑑𝑑𝑒𝑒⃗ 𝑑𝑑𝑒𝑒⃗ ∙ 𝑛𝑛�⃗ 𝑒𝑒⃗ ∙ 𝑑𝑑𝑛𝑛�⃗ 𝑛𝑛�⃗ ∙ 𝑑𝑑𝑛𝑛�⃗

= (37)

𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 𝑖𝑖 where are one-forms on M and 𝑑𝑑 are𝑒𝑒⃗ one𝜔𝜔-forms𝑒𝑒⃗ − 𝜔𝜔on𝑛𝑛� ⃗Rn+1. But + = 0 implies

that 𝜔𝜔𝑖𝑖𝑖𝑖 𝜔𝜔𝑖𝑖 𝑑𝑑𝑒𝑒⃗𝑖𝑖 ∙ 𝑒𝑒⃗𝑘𝑘 𝑒𝑒⃗𝑖𝑖 ∙ 𝑑𝑑𝑒𝑒⃗𝑘𝑘

+ = 0

𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 And + = 0 and = 0𝜔𝜔, imply,𝜔𝜔

𝑑𝑑𝑒𝑒⃗𝑖𝑖 ∙ 𝑛𝑛�⃗ 𝑒𝑒⃗𝑖𝑖 ∙ 𝑑𝑑𝑛𝑛�⃗ 𝑛𝑛�⃗ ∙ 𝑑𝑑𝑛𝑛�⃗ 16

𝑖𝑖 𝑖𝑖 As in our discussion on surfaces in R3, it is 𝑑𝑑convenient𝑛𝑛�⃗ 𝜔𝜔 𝑒𝑒⃗ to define matrices,

1 𝑒𝑒⃗. 2 = ⎛𝑒𝑒⃗. ⎞ , = ( , , … , ) , = ( , , … , ) . ⎜ ⎟ 1 2 𝑛𝑛 1 2 𝑛𝑛 𝑒𝑒⃗ ⎜ . ⎟ 𝜎𝜎⃗ 𝜎𝜎 𝜎𝜎 𝜎𝜎 𝜔𝜔��⃗ 𝜔𝜔 𝜔𝜔 𝜔𝜔 ⎜ ⎟

𝑛𝑛 And, ⎝𝑒𝑒⃗ ⎠

𝑖𝑖𝑖𝑖 Then we can write, Ω �𝜔𝜔 �

= , = , = (38) 0 𝑇𝑇 𝑇𝑇 𝑑𝑑𝑥𝑥⃗ 𝜎𝜎⃗𝑒𝑒⃗ 𝑑𝑑 �𝑒𝑒⃗� � Ω −𝜔𝜔��⃗ � �𝑒𝑒⃗� Ω −Ω Next we can write out the integrability𝑛𝑛 �⃗condition,𝜔𝜔��⃗ 𝑛𝑛�⃗

0 = ( ) = ( ) ( ) = ( ) ( ) = ( ) + ( ) 𝑇𝑇 𝑇𝑇 So, 𝑑𝑑 𝑑𝑑𝑥𝑥⃗ 𝑑𝑑𝜎𝜎⃗ 𝑒𝑒⃗ − 𝜎𝜎⃗ 𝑑𝑑𝑒𝑒⃗ 𝑑𝑑𝜎𝜎⃗ 𝑒𝑒⃗ − 𝜎𝜎⃗ ∧ Ω𝑒𝑒⃗ − 𝜔𝜔��⃗ 𝑛𝑛�⃗ 𝑑𝑑𝜎𝜎⃗ − 𝜎𝜎⃗ ∧ Ω 𝑒𝑒⃗ 𝜎𝜎⃗ ∧ 𝜔𝜔��⃗ 𝑛𝑛�⃗

= , = 0 (39) 𝑇𝑇 Next, 𝑑𝑑𝜎𝜎⃗ 𝜎𝜎⃗ ∧ Ω 𝜎𝜎⃗ ∧ 𝜔𝜔��⃗

0 = = d 0 𝑇𝑇 0 𝑇𝑇 𝑑𝑑 �𝑑𝑑 �𝑒𝑒⃗�� Ω −𝑑𝑑𝜔𝜔��⃗ � �𝑒𝑒⃗� − � Ω −𝜔𝜔��⃗ � 𝑑𝑑 �𝑒𝑒⃗� d 𝑛𝑛�⃗ = 𝑑𝑑𝜔𝜔��⃗ 𝑛𝑛�⃗ 𝜔𝜔��⃗ 𝑛𝑛�⃗ 0 𝑇𝑇 0 𝑇𝑇 2 � Ω −𝑑𝑑𝜔𝜔��⃗ � �𝑒𝑒⃗� − � Ω −𝜔𝜔��⃗ � �𝑒𝑒⃗� d 𝑑𝑑𝜔𝜔�+�⃗ 𝑛𝑛�⃗ +𝜔𝜔��⃗ 𝑛𝑛�⃗ = 𝑇𝑇 𝑇𝑇 𝑇𝑇 � Ω − Ω ∧ Ω 𝜔𝜔��⃗ ∧ 𝜔𝜔 −𝑑𝑑𝜔𝜔��⃗ Ω𝑇𝑇∧ 𝜔𝜔��⃗ � �𝑒𝑒⃗� Collecting, 𝑑𝑑𝑑𝑑 − 𝜔𝜔 ∧ Ω 𝜔𝜔 ∧ 𝜔𝜔��⃗ 𝑛𝑛�⃗

d + = 0 , = (40) 𝑇𝑇 Ω − Ω ∧ Ω 𝜔𝜔��⃗ ∧ 𝜔𝜔 𝑑𝑑𝑑𝑑 𝜔𝜔 ∧ Ω

The combination d occurs frequently and is significant geometrically, as we will see,

so define, Ω − Ω ∧ Ω

= = d (41)

𝑖𝑖𝑖𝑖 which satisfies, Θ �𝜃𝜃 � Ω − Ω ∧ Ω

+ = 0 (42) 𝑇𝑇 Collecting everything in terms of entriesΘ in𝜔𝜔��⃗ the∧ matrix,𝜔𝜔

= , = , = 0 (43)

𝑗𝑗 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 𝑖𝑖 𝑖𝑖 And, 𝑑𝑑𝜎𝜎 𝜎𝜎 ∧ 𝜔𝜔 𝜔𝜔 −𝜔𝜔 𝜎𝜎 ∧ 𝜔𝜔

= , + = 0 (44)

𝑗𝑗 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 Since the set { } forms𝑑𝑑 𝜔𝜔a basis𝜔𝜔 of∧ one𝜔𝜔 -forms𝜃𝜃 on 𝜔𝜔M, ∧we𝜔𝜔 can write,

𝑖𝑖 𝜎𝜎 = (45)

𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 Since = = 0, we learn that 𝜔𝜔 is sy𝑏𝑏mmetric,𝜎𝜎 = . 𝑇𝑇 𝑗𝑗 𝑗𝑗 𝑖𝑖𝑗𝑗 𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 𝜎𝜎 ⃗ Generalizing∧ 𝜔𝜔��⃗ 𝜎𝜎 ∧ from𝜔𝜔 our earlier discussion𝑏𝑏 of surfaces S 𝑏𝑏in R3,𝑏𝑏 we define the mean curvature H and the Gaussian curvature K,

1 = =

𝐻𝐻 𝑏𝑏𝑖𝑖𝑖𝑖 𝐾𝐾 𝑑𝑑𝑑𝑑𝑑𝑑�𝑏𝑏𝑖𝑖𝑖𝑖� It follows from Eq. 45 that the volume ele𝑛𝑛 ment,

… = … = …

𝜔𝜔1 ∧ 𝜔𝜔2 ∧ ∧=𝜔𝜔𝑛𝑛 𝑏𝑏1𝑗𝑗𝜎𝜎𝑗𝑗 ∧ …𝑏𝑏2𝑘𝑘𝜎𝜎𝑘𝑘 ∧ ∧ 𝑏𝑏𝑛𝑛𝑛𝑛𝜎𝜎𝑙𝑙 𝑑𝑑𝑑𝑑𝑑𝑑�𝑏𝑏𝑖𝑖𝑖𝑖�𝜎𝜎1 ∧ 𝜎𝜎2 ∧ ∧ 𝜎𝜎𝑛𝑛 1 2 𝑛𝑛 and K represents the ratio of𝐾𝐾 areas𝜎𝜎 ∧ on𝜎𝜎 ∧M and∧ 𝜎𝜎Sn.

Now consider a vector field on M. is tangent to M, so,

𝑣𝑣⃗ 𝑣𝑣⃗ =

𝑖𝑖 𝑖𝑖 which we can differentiate, 𝑣𝑣⃗ 𝑐𝑐 𝑒𝑒⃗

= + = + = +

𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑖𝑖 where we used𝑑𝑑𝑣𝑣⃗ Eq.𝑑𝑑𝑐𝑐 37.𝑒𝑒⃗ We𝑐𝑐 𝑑𝑑le𝑒𝑒⃗arn that𝑑𝑑𝑐𝑐 𝑒𝑒if⃗ does𝑐𝑐 �𝜔𝜔 not𝑒𝑒⃗ change− 𝜔𝜔 𝑛𝑛�⃗ �on the�𝑑𝑑 𝑐𝑐surface,𝑐𝑐 𝜔𝜔 then�𝑒𝑒⃗ − 𝑐𝑐 𝜔𝜔 𝑛𝑛�⃗

𝑣𝑣⃗ + = 0 (46)

𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 If this is true for the vector field , we𝑑𝑑 say𝑐𝑐 that𝑐𝑐 𝜔𝜔 moves by “parallel transport” along the surface.

This is the same concept discussed𝑣𝑣⃗ at length in 𝑣𝑣the⃗ textbook and in several Supplementary Lectures, especially #9. Using the same analysis as in our discussion of surfaces in R3, we see that two vector fields, and , which move by parallel transport along M, have a constant inner

product, . 𝑣𝑣⃗ 𝑤𝑤��⃗

In𝑣𝑣⃗ addition,∙ 𝑤𝑤��⃗ suppose that ( ) is a curve on M, where s is the curve’s arc-length, and

( ) = is the curve’s unit𝑃𝑃� ⃗tangent,𝑠𝑠 then the curve is a geodesic if ( ) moves by parallel 𝑡𝑡transport.⃗ 𝑠𝑠 𝑑𝑑𝑃𝑃� ⃗We⁄𝑑𝑑𝑑𝑑 discussed the geometry underlying this terminology in Supplementary𝑡𝑡⃗ 𝑠𝑠 Lecture 9: if the tangent moves by parallel transport, then the curve is as straight as possible on the hypersurface M.

Now consider the curvature matrix . Its matrix elements are two-forms so they can be written as linear superpositions of , Θ . The coefficients make𝜃𝜃𝑖𝑖𝑖𝑖 up the famous Riemann curvature tensor, �𝜎𝜎𝑖𝑖 𝜎𝜎𝑗𝑗�

= , + = 0 (47) 1 𝜃𝜃𝑖𝑖𝑖𝑖 2 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝜎𝜎𝑘𝑘 ∧ 𝜎𝜎𝑙𝑙 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

We will check later that is the Riemann curvature tensor introduced in the textbook and

Supplementary Lectures 𝑅𝑅in𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖terms of parallel transport and holonomy. This will be done after introducing Christoffel connections.

Back to the business at hand. We had the defining relations, Eq. 44,

+ = 0

𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 So, we can relate this to Eq. 47 using the𝜃𝜃 fact,𝜔𝜔 Eq.∧ 45,𝜔𝜔 that = ,

𝜔𝜔𝑖𝑖 𝑏𝑏𝑖𝑖𝑖𝑖𝜎𝜎𝑗𝑗 19

1 = = 2 𝜔𝜔𝑖𝑖 ∧ 𝜔𝜔𝑗𝑗 𝑏𝑏𝑖𝑖𝑘𝑘𝑏𝑏𝑗𝑗𝑗𝑗 𝜎𝜎𝑘𝑘 ∧ 𝜎𝜎𝑙𝑙 �𝑏𝑏𝑖𝑖𝑘𝑘𝑏𝑏𝑗𝑗𝑗𝑗 − 𝑏𝑏𝑖𝑖𝑖𝑖𝑏𝑏𝑗𝑗𝑗𝑗�𝜎𝜎𝑘𝑘 ∧ 𝜎𝜎𝑙𝑙 So the Riemann curvature tensor satisfies,

+ = 0 (48) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏 𝑏𝑏 𝑅𝑅 𝑑𝑑𝑑𝑑𝑑𝑑 � 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗� This result gives the symmetry relations of , 𝑏𝑏 𝑏𝑏

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + =𝑅𝑅0 , + = 0

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 𝑅𝑅 + +𝑅𝑅 =𝑅𝑅0

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑅𝑅 𝑅𝑅 = 𝑅𝑅

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 Now let’s briefly discuss the equations𝑅𝑅 , 𝑅𝑅

= , + = 0 𝑇𝑇 We will see in the next section that they𝑑𝑑𝜎𝜎⃗ determine𝜎𝜎⃗ ∧ Ω Ω. ThisΩ means that is determined by

alone. Given this result, it will follow that, Ω Ω 𝜎𝜎⃗

= d

is also completely determined by . In otherΘ words,Ω − Ω the∧ ΩRiemann curvature tensor is an intrinsic

property of the surface S. More below.𝜎𝜎⃗

Intrinsic Geometry of Manifolds Now let’s ask what aspects of the geometry of the manifold M exist independent of any Euclidean space it might be embedded in [1]. In this case we must postulate the existence of a local geometry, we cannot inherit it from Rn+1. Of course, the reason we discussed manifolds embedded in Rn+1 was to motivate this step and to make the best definitions!

So, let M be a n-dimensional manifold. The manifold is endowed with an inner product, so if and are two tangent vectors, then is a given, smooth real function. Suppose that { } there is𝑣𝑣⃗ a basis𝑤𝑤��⃗ , , … , which is orthonormal,𝑣𝑣⃗ ∙ 𝑤𝑤��⃗ = . If P is a point on M, then we postulate, 𝑒𝑒⃗1 𝑒𝑒⃗2 𝑒𝑒⃗𝑛𝑛 𝑒𝑒⃗𝑖𝑖 ∙ 𝑒𝑒⃗𝑗𝑗 𝛿𝛿𝑖𝑖𝑖𝑖

= (49)

𝑖𝑖 𝑖𝑖 where the differential operator will 𝒅𝒅prove𝑃𝑃 𝜎𝜎to𝑒𝑒 ⃗be the covariant derivative introduced earlier in

the context of classical differential𝒅𝒅 geometry (Some authors use the notation D here, but we will follow the conventions of [1] and use a bold face . See the appendix to this lecture for more.) If { } there is a coordinate system , , … , on M,𝒅𝒅 then, 1 2 𝑛𝑛 𝑢𝑢 𝑢𝑢 𝑢𝑢 = 𝑖𝑖 𝜕𝜕 𝒅𝒅𝑃𝑃 𝑑𝑑𝑢𝑢 𝑖𝑖 and , , … , is a natural frame associated with𝜕𝜕𝑢𝑢 the coordinate system. A standard 𝜕𝜕 𝜕𝜕 𝜕𝜕 1 2 𝑛𝑛 application�𝜕𝜕𝑢𝑢 𝜕𝜕𝑢𝑢 of the 𝜕𝜕chain𝑢𝑢 � rule shows that is independent of the choice of a particular coordinate system. 𝒅𝒅𝑃𝑃

Now we have to potulate how changes under a displacement . In this case there is no

“normal direction”, so we postulate, 𝑒𝑒⃗𝑖𝑖 𝒅𝒅

= (50)

𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 The inspiration for this postulate, the 𝒅𝒅defining𝑒𝑒⃗ 𝜔𝜔 conditi𝑒𝑒⃗ on for , is Eq. 37. If we were in Rn+1,we n+1 would describe as the projection of the differential in R 𝒅𝒅onto the tangent plane, which is, in fact, the definition𝒅𝒅 of the covariant derivative. With this understanding, much of the developments here can be borrowed from our earlier discussions of hypersurfaces in Rn+1.

Now we need to find the one-forms which are consistent with the integrability

conditions, 𝜔𝜔𝑖𝑖𝑖𝑖

+ = 0 , ( ) = 0

𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑘𝑘 The first condition implies 𝒅𝒅𝑒𝑒⃗+∙ 𝑒𝑒⃗ = 𝑒𝑒⃗0, ∙as𝒅𝒅 𝑒𝑒⃗before. The𝒅𝒅 second𝒅𝒅𝑃𝑃 condition reads,

𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 𝜔𝜔 𝜔𝜔 ( ) = 0

𝑖𝑖 𝑖𝑖 So, 𝒅𝒅 𝜎𝜎 𝑒𝑒⃗

= = 0

𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 which implies, 𝒅𝒅𝜎𝜎 𝑒𝑒⃗ − 𝜎𝜎 𝒅𝒅𝑒𝑒⃗ �𝒅𝒅𝜎𝜎 − 𝜎𝜎 ∧ 𝜔𝜔 �𝑒𝑒⃗

= (51)

𝑖𝑖 𝑗𝑗 𝑗𝑗𝑗𝑗 So, our problem at hand is to solve 𝒅𝒅𝜎𝜎= 𝜎𝜎 ∧ 𝜔𝜔 subject to the condition + = 0. First,

{ } is a complete set of one-forms,𝒅𝒅 so𝜎𝜎𝑖𝑖 there𝜎𝜎𝑗𝑗 must∧ 𝜔𝜔𝑗𝑗𝑗𝑗 be coefficients, called connection𝜔𝜔𝑖𝑖𝑖𝑖 𝜔𝜔𝑗𝑗𝑗𝑗 coefficients 𝜎𝜎𝑘𝑘 , or Christoffel symbols, 𝑖𝑖𝑖𝑖𝑖𝑖 Γ =

𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 We will see later that coincide with the𝜔𝜔 ChristoffelΓ 𝜎𝜎 symbols introduced earlier in the

textbook and SupplementaryΓ𝑖𝑖𝑖𝑖𝑖𝑖 Lectures. The anti-symmetry of implies, 𝑖𝑖𝑖𝑖 + = 0 𝜔𝜔

𝑖𝑖𝑖𝑖𝑘𝑘 𝑗𝑗𝑗𝑗𝑗𝑗 But the are known because the are Γknown.Γ We can write out completeness again,

𝒅𝒅𝜎𝜎𝑖𝑖 𝜎𝜎1𝑖𝑖 = , + = 0 2 𝒅𝒅𝜎𝜎𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝜎𝜎𝑗𝑗 ∧ 𝜎𝜎𝑘𝑘 𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖 Then,

1 1 = = = = 2 2 𝒅𝒅𝜎𝜎𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝜎𝜎𝑗𝑗 ∧ 𝜎𝜎𝑘𝑘 𝜎𝜎𝑗𝑗 ∧ 𝜔𝜔𝑗𝑗𝑗𝑗 Γ𝑗𝑗𝑗𝑗𝑗𝑗 𝜎𝜎𝑗𝑗 ∧ 𝜎𝜎𝑘𝑘 �Γ𝑗𝑗𝑗𝑗𝑗𝑗 − Γ𝑘𝑘𝑘𝑘𝑘𝑘�𝜎𝜎𝑗𝑗 ∧ 𝜎𝜎𝑘𝑘 So,

𝑗𝑗𝑗𝑗𝑗𝑗 𝑘𝑘𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖𝑖𝑖 We can solve this system of equations forΓ − Γby writing𝑐𝑐 down the two other cyclic permutations of the indices ( ), adding theΓ𝑖𝑖𝑖𝑖 𝑖𝑖first two and subtracting the third to isolate . The result is, 𝑗𝑗𝑗𝑗𝑗𝑗 → 𝑘𝑘𝑘𝑘𝑘𝑘 → 𝑖𝑖𝑖𝑖𝑖𝑖 Γ𝑘𝑘𝑘𝑘𝑘𝑘 1 = + 2 Γ𝑖𝑖𝑖𝑖𝑖𝑖 �𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗𝑗𝑗𝑗𝑗 − 𝑐𝑐𝑘𝑘𝑘𝑘𝑘𝑘� All of this will become more familiar when we turn to metric spaces below.

Let’s derive a further integrability condition by calculating the exterior derivative of = . Now,

𝑑𝑑𝜎𝜎⃗ 𝜎𝜎⃗0∧=Ω ( ) = = ( ) = ( d )

𝑑𝑑 𝑑𝑑𝜎𝜎⃗ 𝑑𝑑𝜎𝜎⃗ ∧ Ω − 𝜎𝜎⃗ ∧ 𝑑𝑑Ω 𝜎𝜎⃗ ∧ Ω ∧ Ω − 𝜎𝜎⃗ ∧ 𝑑𝑑Ω 𝜎𝜎⃗ ∧ Ω ∧ Ω − Ω 22

Therefore,

= 0 where is the curvature form introduced earlier𝜎𝜎⃗ ∧ Θ,

Θ = d

The exterior derivative of is Θinformative,Ω − Ω ∧ Ω

= d(d ) Θ ( ) = 0 (d ) + (d )

𝑑𝑑 Θ = ( Ω+− 𝑑𝑑 Ω)∧ Ω + − ( Ω+∧ Ω )Ω=∧ Ω

which comprise the “Bianchi− identities”Θ Ω ∧ Ω which∧ Ω wereΩ introduced∧ Θ Ω ∧ brieflyΩ inΩ the∧ Θ textbook.− Θ ∧ Ω

Finally, we should consider the rate of change of ,

𝑖𝑖 = ( ) = ( ) = 𝒅𝒅𝑒𝑒⃗ = (d ) = 𝟐𝟐 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 where we 𝒅𝒅have𝑒𝑒⃗ identified𝒅𝒅 𝒅𝒅𝑒𝑒⃗ the𝒅𝒅 curvatureΩ ∧ 𝑒𝑒⃗ form𝑑𝑑Ω again.∧ 𝑒𝑒⃗ − So,Ω ∧ the𝒅𝒅𝑒𝑒 ⃗“acceleration”Ω − Ω ∧ ofΩ the𝑒𝑒⃗ frameΘ𝑒𝑒⃗ is proportional to the curvature.

As discussed in Eq. 47, the two-forms which make up the curvature matrix, =

, can be expanded in terms of the basis 𝜃𝜃𝑖𝑖𝑖𝑖 . The coefficients of the completenessΘ �statement𝜃𝜃𝑖𝑖𝑖𝑖� will be identified as the componen�ts𝜎𝜎 𝑖𝑖of∧ the𝜎𝜎𝑗𝑗� Riemann tensor, , 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = (47) 1 𝜃𝜃𝑖𝑖𝑖𝑖 2 𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝜎𝜎𝑘𝑘 ∧ 𝜎𝜎𝑙𝑙 Several of the important symmetries of follow directly from this equation,

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑅𝑅 + = 0 , + = 0 (52.a)

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑗𝑗𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 In addition, the integrability condition𝑅𝑅 =𝑅𝑅 0 becomes, 𝑅𝑅 𝑅𝑅

𝜎𝜎⃗ ∧ Θ = 0

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑘𝑘 which implies that, 𝑅𝑅 𝜎𝜎 ∧ 𝜎𝜎 ∧ 𝜎𝜎

+ + = 0 (52.b)

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 The final symmetry, 𝑅𝑅 𝑅𝑅 𝑅𝑅

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 follows from combining Eq. 52.a and 52.b.𝑅𝑅 𝑅𝑅

Now let’s introduce the metric. We have the natural frame = , … , = . 𝜕𝜕 𝜕𝜕 1 𝑛𝑛 Then the entries in the metric tensor are defined to be the inner products�𝑣𝑣⃗1 , 𝜕𝜕𝑢𝑢 𝑣𝑣⃗𝑛𝑛 𝜕𝜕𝑢𝑢 �

𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 so the invariant distance element is, 𝑔𝑔 𝑣𝑣⃗ ∙ 𝑣𝑣⃗

= = = = 2 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 Next we write 𝑑𝑑𝑑𝑑 in terms𝒅𝒅𝑃𝑃 ∙ 𝒅𝒅 of𝑃𝑃 the� complete𝑣𝑣⃗ 𝑑𝑑𝑑𝑑 � ∙ � set𝑣𝑣⃗ 𝑑𝑑𝑑𝑑 { �}, 𝑣𝑣⃗ ∙ 𝑣𝑣⃗ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑔𝑔 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

𝒅𝒅𝑣𝑣⃗𝑖𝑖 𝑣𝑣⃗𝑖𝑖 = 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝒅𝒅𝑣𝑣⃗ 𝜂𝜂𝑖𝑖 𝑣𝑣⃗ where are one-forms which can be written as linear superpositions of 𝑗𝑗 𝑗𝑗 �𝜂𝜂𝑖𝑖 � �𝑑𝑑𝑑𝑑 � = 𝑗𝑗 𝑗𝑗 𝑘𝑘 𝜂𝜂𝑖𝑖 Γ 𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑 Now let’s find in terms of the metric and its derivatives to show that is indeed the 𝑗𝑗 𝑗𝑗 Christoffel symbolsΓ𝑖𝑖𝑖𝑖 introduced earlier. First, differentiating = ,Γ 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 = + 𝑔𝑔 𝑣𝑣⃗ ∙ 𝑣𝑣⃗

𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 We have, 𝑑𝑑𝑔𝑔 𝒅𝒅𝑣𝑣⃗ ∙ 𝑣𝑣⃗ 𝑣𝑣⃗ ∙ 𝒅𝒅𝑣𝑣⃗

= + = + = + 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑑𝑑𝑔𝑔𝑖𝑖𝑖𝑖 𝜂𝜂𝑖𝑖 𝑣𝑣⃗𝑘𝑘 ∙ 𝑣𝑣⃗𝑗𝑗 𝑣𝑣⃗𝑖𝑖 ∙ 𝜂𝜂𝑗𝑗 𝑣𝑣⃗𝑘𝑘 𝜂𝜂𝑖𝑖 𝑔𝑔𝑘𝑘𝑘𝑘 𝜂𝜂𝑗𝑗 𝑔𝑔𝑖𝑖𝑖𝑖 Γ 𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑 𝑔𝑔𝑘𝑘𝑘𝑘 Γ 𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑 𝑔𝑔𝑖𝑖𝑖𝑖 = + 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖 So, we learn, 𝑑𝑑𝑔𝑔 �Γ 𝑔𝑔 Γ 𝑔𝑔 �𝑑𝑑𝑢𝑢

= + (53) 𝜕𝜕𝑔𝑔𝑖𝑖𝑖𝑖 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝜕𝜕𝑢𝑢 Γ 𝑖𝑖𝑖𝑖𝑔𝑔𝑘𝑘𝑘𝑘 Γ 𝑗𝑗𝑗𝑗𝑔𝑔𝑖𝑖𝑖𝑖 But requiring,

0 = ( ) = 𝑖𝑖 𝒅𝒅 𝒅𝒅𝑃𝑃 −𝑑𝑑𝑑𝑑 𝒅𝒅𝑣𝑣⃗𝑖𝑖 24

we learn a symmetry property of the , 𝑘𝑘 Γ𝑖𝑖𝑖𝑖 0 = = = 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑗𝑗 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 𝑗𝑗 which implies, −𝑑𝑑𝑑𝑑 𝒅𝒅𝑣𝑣⃗ −𝑑𝑑𝑑𝑑 𝜂𝜂 𝑣𝑣⃗ Γ 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝑣𝑣⃗

= 𝑗𝑗 𝑗𝑗 𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘 We lower indices with Γ Γ

= 𝑙𝑙 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 So Eq. 53 becomes, Γ 𝑔𝑔 Γ

= + (54) 𝜕𝜕𝑔𝑔𝑖𝑖𝑖𝑖 𝑙𝑙 𝜕𝜕𝑢𝑢 Γ𝑗𝑗𝑗𝑗𝑗𝑗 Γ𝑖𝑖𝑖𝑖𝑙𝑙 We have met this set of equations before and know how to solve it for a particular : We write

the equation and then obtain two more by cyclically permuting twice. Taking theΓ𝑖𝑖𝑖𝑖 sum𝑖𝑖 of the first two and subtracting the third isolates a single and we find,𝑙𝑙𝑙𝑙𝑙𝑙 𝑖𝑖𝑖𝑖𝑖𝑖 1 Γ = + 2 Γ𝑖𝑖𝑖𝑖𝑖𝑖 �𝜕𝜕𝑘𝑘𝑔𝑔𝑖𝑖𝑖𝑖 𝜕𝜕𝑗𝑗𝑔𝑔𝑖𝑖𝑖𝑖 − 𝜕𝜕𝑖𝑖𝑔𝑔𝑗𝑗𝑗𝑗� where = . This is precisely the formula for the Christoffel symbols that we derived in 𝑘𝑘 the tensor𝜕𝜕𝑘𝑘 analysis𝜕𝜕⁄𝜕𝜕𝑢𝑢 section in the textbook.

Next, let’s discuss parallel transport in this scenario. If is a tangent vector in M, then

= 𝑣𝑣⃗

𝑖𝑖 𝑖𝑖 where will depend on the position P. The derivative𝑣𝑣⃗ 𝑐𝑐 𝑒𝑒⃗ of is,

𝑖𝑖 𝑐𝑐 = + = + 𝑣𝑣⃗= +

𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 So, if moves by parallel𝒅𝒅�𝒗𝒗�⃗ transpor𝑑𝑑𝑐𝑐 𝑒𝑒⃗ t𝒅𝒅, meaning𝑒𝑒⃗ 𝑑𝑑𝑐𝑐 ,𝑒𝑒 ⃗ =𝑐𝑐 𝜔𝜔0, then𝑒𝑒⃗ �𝑑𝑑𝑐𝑐 𝑐𝑐 𝜔𝜔 �𝑒𝑒⃗

𝑣𝑣⃗ 𝒅𝒅�𝒗𝒗�⃗ + = 0 , + = 0 (55) 𝜕𝜕𝑐𝑐𝑗𝑗 𝑘𝑘 𝑑𝑑𝑐𝑐𝑖𝑖 𝑐𝑐𝑖𝑖𝜔𝜔𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝜕𝜕𝑢𝑢 Γ𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑖𝑖 which is in fact the formula for parallel transport derived in the textbook.

Non-Euclidean Geometry Let’s apply this formalism [1] to the geometry of the Poincare Upper-Half Plane model of a space with a constant negative Gaussian curvature, = 1. This topic was discussed in

Supplementary Lecture 4 using complex analysis. 𝐾𝐾 −

Recall the metric for the upper half plane, ( 0, ), model

𝑦𝑦 ≥ ∞ ≥ 𝑥𝑥 ≥ −∞ = 2 2 (56.a) 2 𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 𝑦𝑦 So,

= , = 0 , = (56.b) −2 −2 11 12 22 An orthonormal basis for the space𝑔𝑔 will𝑦𝑦 be, 𝑔𝑔 𝑔𝑔 𝑦𝑦

= ( , 0) and = (0, ) (57)

1 2 Let’s check that = .First,𝑒𝑒⃗ 𝑦𝑦 𝑒𝑒⃗ 𝑦𝑦

𝑖𝑖 𝑗𝑗 𝑖𝑖𝑖𝑖 𝑒𝑒⃗ ∙ 𝑒𝑒⃗ 𝛿𝛿 = (1,0) (1,0) = = 1 2 2 1 1 11 And similarly = 0 and𝑒𝑒 ⃗ ∙ 𝑒𝑒⃗ =𝑦𝑦1. ∙ 𝑦𝑦 𝑔𝑔

1 2 2 2 If we write,𝑒𝑒⃗ ∙ 𝑒𝑒⃗ 𝑒𝑒⃗ ∙ 𝑒𝑒⃗

= ( , ) =

𝑖𝑖 𝑖𝑖 then, 𝒅𝒅𝑃𝑃 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜎𝜎 𝑒𝑒⃗

= and = (58) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜎𝜎1 𝑦𝑦 𝜎𝜎2 𝑦𝑦 Now we can start investigating the geometry of this space. The integrability conditions read,

1 1 = = = , = 0

𝑑𝑑𝜎𝜎1 − 2 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝜎𝜎1 ∧ 𝜎𝜎2 𝑑𝑑𝜎𝜎2 We can identify the matric 𝑦𝑦 from its definition,𝑦𝑦

Ω =

𝑑𝑑𝜎𝜎⃗ 𝜎𝜎⃗ ∧ Ω 26

Explicitly,

( , ) = ( , )

1 2 1 2 Or, 𝑑𝑑𝜎𝜎 𝑑𝑑𝜎𝜎 𝜎𝜎 𝜎𝜎 ∧ Ω

( , 0) = ( , )

1 2 1 2 So, we infer, 𝜎𝜎 ∧ 𝜎𝜎 𝜎𝜎 𝜎𝜎 ∧ Ω

0 = = 0 1 𝑖𝑖𝑖𝑖 𝜎𝜎 Ω �𝜔𝜔 � � 1 � using the fact that is anti-symmetric to infer −=𝜎𝜎 .

12 1 Now we canΩ calculate the curvature matrix,ω 𝜎𝜎

0 0 0 0 1 = d = = 0 0 0 1 0 𝑑𝑑𝑑𝑑1 𝜎𝜎1 𝜎𝜎1 Θ Ω − Ω ∧ Ω 0� 1 � − � � ∧ � � � � 𝑑𝑑𝑑𝑑1 = −𝑑𝑑𝑑𝑑1 −𝜎𝜎1 −𝜎𝜎1 1 0 − � � 𝜎𝜎1 ∧ 𝜎𝜎2 But recall the definition −= . So, we read off 1 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 𝑙𝑙 𝜃𝜃 1 2 𝑅𝑅 𝜎𝜎 ∧ 𝜎𝜎 1 = + = 2 2 𝜃𝜃12 𝑅𝑅1212 𝜎𝜎1 ∧ 𝜎𝜎2 𝑅𝑅1221 𝜎𝜎2 ∧ 𝜎𝜎1 𝑅𝑅1212 𝜎𝜎1 ∧ 𝜎𝜎2 So, = 1. Now we can obtain the Gaussian intrinsic curvature K using the integrability

condition𝑅𝑅1212 + = 0 and = = . So, = = which implies = 1. We learn𝑑𝑑𝜔𝜔� that𝐾𝐾 𝜎𝜎1 =∧ 𝜎𝜎21 = 𝜔𝜔� and𝜔𝜔12 the space𝜎𝜎1 has𝑑𝑑 𝜔𝜔�a constant𝑑𝑑𝜎𝜎1 negative𝜎𝜎1 ∧ 𝜎𝜎2 curvature, as 𝐾𝐾 −claimed. 𝐾𝐾 − −𝑅𝑅1212 Finally let’s check that semi-circles with their centers on the real axis are geodesics in the upper-half plane. We discussed this in Supplementary Lecture 4 using complex analysis and it would be instructive to see how it works out here. In ( , ) coordinates, the geodesics read,

= + cos , 𝑥𝑥 =𝑦𝑦 sin

for 0 . We need the tangent𝑥𝑥 𝑎𝑎to 𝑟𝑟= 𝜑𝜑( ), (𝑦𝑦 ) ,𝑟𝑟 𝜑𝜑

≤ 𝜑𝜑 ≤ 𝜋𝜋 𝑃𝑃 �𝑥𝑥 𝜑𝜑 𝑦𝑦 𝜑𝜑 � 1 = ( sin , cos ) = [( sin ) + (cos ) ] = [( sin ) + (cos ) ] sin 𝒅𝒅𝑃𝑃 𝑟𝑟 𝑟𝑟 − 𝜑𝜑 𝜑𝜑 − 𝜑𝜑 𝑒𝑒⃗1 𝜑𝜑 𝑒𝑒⃗2 − 𝜑𝜑 𝑒𝑒⃗1 𝜑𝜑 𝑒𝑒⃗2 𝑑𝑑𝑑𝑑 𝑦𝑦 27 𝜑𝜑

But along the curve,

+ ( sin + cos )( ) ( ) = = = 2 2 2 2 sin2 2 2 sin 2 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟 𝜑𝜑 𝑟𝑟 𝜑𝜑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 2 2 2 So, the unit tangent vector is, 𝑦𝑦 𝑟𝑟 𝜑𝜑 𝜑𝜑

= = = ( sin ) + (cos ) 𝒅𝒅𝑃𝑃 𝒅𝒅𝑃𝑃 𝑑𝑑𝑑𝑑 𝑡𝑡⃗ − 𝜑𝜑 𝑒𝑒⃗1 𝜑𝜑 𝑒𝑒⃗2 But along the curve, 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

sin = = = = = sin sin 𝑑𝑑𝑑𝑑 −𝑟𝑟 𝜑𝜑 𝑑𝑑𝑑𝑑 𝜔𝜔12 𝜎𝜎1 −𝑑𝑑𝑑𝑑 − 𝜑𝜑𝜑𝜑𝜑𝜑 And, 𝑦𝑦 𝑟𝑟 𝜑𝜑

0 = = 0 𝜔𝜔12 𝑒𝑒⃗1 𝒅𝒅𝑒𝑒⃗ Ω𝑒𝑒⃗ � 12 � � 2� So, −𝜔𝜔 𝑒𝑒⃗

= =

1 2 2 1 And finally, 𝒅𝒅𝑒𝑒⃗ −𝑑𝑑𝑑𝑑 𝑒𝑒⃗ 𝒅𝒅𝑒𝑒⃗ 𝑑𝑑𝑑𝑑 𝑒𝑒⃗

= [( sin ) + (cos ) ]

1 2 ( ) ( ) 𝒅𝒅𝒕𝒕⃗ 𝒅𝒅 − =𝜑𝜑 𝑒𝑒⃗ cos 𝜑𝜑 𝑒𝑒⃗ sin sin + cos = 0 1 2 2 1 So, the unit tangent moves− by𝜑𝜑 parallel𝑑𝑑𝑑𝑑 𝑒𝑒⃗ − transport𝜑𝜑 𝑑𝑑𝑑𝑑 and𝑒𝑒⃗ −the curve,𝜑𝜑 − 𝑑𝑑the𝑑𝑑𝑒𝑒 ⃗semi-circle𝜑𝜑 with𝑑𝑑𝑑𝑑 𝑒𝑒 ⃗center on

the real axis, is indeed𝒕𝒕⃗ a geodesic.

Appendix: Making Contact with Classical Differential and Riemannian Geometry The presentation of differential forms in this lecture has been very brief because many of the concepts and formulas are already familiar from past discussions of classical differential geometry and tensor analysis. Let’s make the relations more explicit.

Consider a hypersurface in Rn+1. Then points on the n-dimensional surface S are located with a vector ( , … , ), using for a smooth, singularity-free n-dimensional mesh. The 1 𝑛𝑛 𝑖𝑖 tangent space 𝑅𝑅�at⃗ 𝑢𝑢 is spanned𝑢𝑢 by vectors�𝑢𝑢 � = . The partial derivatives with respect to the 𝑖𝑖 mesh coordinates𝑅𝑅�⃗ produce vectors spanned𝑒𝑒⃗𝑖𝑖 by 𝜕𝜕{𝑅𝑅�⃗⁄} 𝜕𝜕and𝑢𝑢 , the unit normal to S at . Using notation that follows the conventions in the textbook,𝑒𝑒⃗𝑖𝑖 𝑛𝑛�⃗ 𝑅𝑅�⃗

= + (A.1) 𝑘𝑘 𝑗𝑗 𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 where = , are the Christoffel𝜕𝜕 𝑒𝑒⃗ connections𝑒𝑒⃗ Γ 𝑛𝑛�⃗ 𝑏𝑏and will be identified as the second 𝑖𝑖 𝑘𝑘 fundamental𝜕𝜕𝑗𝑗 𝜕𝜕 form⁄𝜕𝜕𝑢𝑢 ofΓ 𝑖𝑖𝑖𝑖the hyperspace S. We can easily see that𝑏𝑏𝑖𝑖𝑗𝑗 and are symmetric in their 𝑘𝑘 indices i and j. This follows from the identity, an integrability Γcondition,𝑖𝑖𝑖𝑖 𝑏𝑏𝑖𝑖𝑖𝑖

= = = 2 2 𝜕𝜕 𝑅𝑅�⃗ 𝜕𝜕 𝑅𝑅�⃗ 𝜕𝜕𝑗𝑗𝑒𝑒⃗𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝜕𝜕𝑖𝑖𝑒𝑒⃗𝑗𝑗 So, 𝜕𝜕𝑢𝑢 𝜕𝜕𝑢𝑢 𝜕𝜕𝑢𝑢 𝜕𝜕𝑢𝑢

= = + = + 𝑘𝑘 𝑘𝑘 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 which proves the point. 𝜕𝜕 𝑒𝑒⃗ 𝜕𝜕 𝑒𝑒⃗ 𝑒𝑒⃗ Γ 𝑛𝑛�⃗𝑏𝑏 𝑒𝑒⃗ Γ 𝑛𝑛�⃗𝑏𝑏

Next we need to make definitions to help identify intrinsic properties of S. An observer on S would measure the terms in but the variation normal to the surface would be 𝑘𝑘 out of his sight. So, we introduce𝑒𝑒⃗𝑘𝑘Γ𝑖𝑖𝑖𝑖 the Covariant𝜕𝜕𝑗𝑗𝑒𝑒⃗𝑖𝑖 Derivative to pick out the terms in Eq.𝑛𝑛�⃗𝑏𝑏𝑖𝑖𝑖𝑖 A.1 lying in the tangent space,

= = (A.2) 𝐷𝐷𝑒𝑒⃗𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑗𝑗 𝑗𝑗 𝐷𝐷𝑒𝑒⃗𝑖𝑖 𝜕𝜕𝑢𝑢 𝑑𝑑𝑢𝑢 𝑒𝑒⃗𝑘𝑘Γ𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 The result is a differential form and is the exterior derivative. We define applied to a scalar function to be the ordinary differential,𝑑𝑑 = . And when applies to a𝐷𝐷 vector, we define, 𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑 𝐷𝐷 = = + ( ) = + = + 𝑖𝑖 𝑘𝑘 (A.3) 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝜕𝜕𝑣𝑣 𝑗𝑗 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝜕𝜕𝑣𝑣 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝐷𝐷𝑣𝑣⃗ 𝐷𝐷�𝑒𝑒⃗𝑖𝑖𝑣𝑣 � 𝑒𝑒⃗𝑖𝑖 𝑑𝑑𝑣𝑣 𝐷𝐷𝑒𝑒⃗𝑖𝑖 𝑣𝑣 𝑒𝑒⃗𝑖𝑖 𝜕𝜕𝑢𝑢 𝑑𝑑𝑢𝑢 𝑒𝑒⃗𝑘𝑘Γ𝑖𝑖𝑖𝑖𝑣𝑣 𝑑𝑑𝑢𝑢 𝑒𝑒⃗𝑘𝑘 �𝜕𝜕𝑢𝑢 Γ𝑖𝑖𝑖𝑖𝑣𝑣 � 𝑑𝑑𝑢𝑢 which is a vector-valued differential form. We learn that the components of the Covariant Derivative are,

= + 𝑘𝑘 𝑘𝑘 (A.4) 𝐷𝐷𝑣𝑣 𝜕𝜕𝑣𝑣 𝑘𝑘 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝜕𝜕𝑢𝑢 𝜕𝜕𝑢𝑢 Γ𝑖𝑖𝑖𝑖𝑣𝑣 which indeed is the covariant derivative introduced in the textbook discussion of tensor analysis.

To see that we can use D to formulate the curvature of S, consider the second covariant derivative of . Here we must consistently use the definition of the exterior derivative ,

𝑖𝑖 𝑒𝑒⃗ = ( ) = = + ( ) 𝑑𝑑 2 𝑘𝑘 𝑗𝑗 𝑘𝑘 𝑗𝑗 𝑘𝑘 𝑗𝑗 𝐷𝐷 𝑒𝑒⃗𝑖𝑖 𝐷𝐷 𝐷𝐷𝑒𝑒⃗𝑖𝑖 𝐷𝐷�𝑒𝑒⃗𝑘𝑘Γ𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 � 𝑒𝑒⃗𝑘𝑘𝑑𝑑�Γ𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 � 𝐷𝐷𝑒𝑒⃗𝑘𝑘 Γ𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 = 𝑙𝑙 + 𝜕𝜕Γ𝑖𝑖𝑖𝑖 𝑚𝑚 𝑗𝑗 𝑙𝑙 𝑚𝑚 𝑘𝑘 𝑗𝑗 𝑒𝑒⃗𝑙𝑙 𝑚𝑚 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝑒𝑒⃗𝑙𝑙Γ𝑘𝑘𝑘𝑘𝑑𝑑𝑢𝑢 ∧ Γ𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 Collecting terms, 𝜕𝜕𝑢𝑢

= + = 𝑙𝑙 𝑙𝑙 (A.5) 𝑖𝑖𝑖𝑖 2 1 𝜕𝜕Γ 𝜕𝜕Γ𝑖𝑖𝑖𝑖 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑚𝑚 𝑗𝑗 𝑙𝑙 𝑚𝑚 𝑗𝑗 𝐷𝐷 𝑒𝑒⃗𝑖𝑖 2 𝑒𝑒⃗𝑙𝑙 �𝜕𝜕𝑢𝑢 − 𝜕𝜕𝑢𝑢 Γ𝑘𝑘𝑘𝑘Γ𝑖𝑖𝑖𝑖 − Γ𝑘𝑘𝑘𝑘Γ𝑖𝑖𝑖𝑖� 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 𝑒𝑒⃗𝑙𝑙θ𝑖𝑖 where we identified the curvature two-form and the explicit formula for the Riemann curvature tensor,

= (A.6a) 𝑙𝑙 𝑙𝑙 𝑚𝑚 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑚𝑚𝑚𝑚 with θ 𝑅𝑅 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑

= + 𝑙𝑙 𝑙𝑙 (A.6b) 𝑖𝑖𝑖𝑖 𝑙𝑙 𝜕𝜕Γ 𝜕𝜕Γ𝑖𝑖𝑖𝑖 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑚𝑚 𝑗𝑗 𝑅𝑅𝑖𝑖 𝑚𝑚𝑚𝑚 𝜕𝜕𝑢𝑢 − 𝜕𝜕𝑢𝑢 Γ𝑘𝑘𝑘𝑘Γ𝑖𝑖𝑖𝑖 − Γ𝑘𝑘𝑘𝑘Γ𝑖𝑖𝑖𝑖 Let’s make closer contact with the notation in the body of this lecture. We have written Eq. 50,

= 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 So we identify, in the notation of standard 𝐷𝐷tensor𝑒𝑒⃗ 𝑒𝑒analysis⃗ 𝜔𝜔 in this appendix,

= (A.7) 𝑘𝑘 𝑘𝑘 𝑗𝑗 𝑖𝑖 𝑖𝑖𝑖𝑖 So, 𝜔𝜔 Γ 𝑑𝑑𝑢𝑢

= = + ( ) = + = + 2 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑙𝑙 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑙𝑙 𝑘𝑘 𝑖𝑖 𝑙𝑙 𝑖𝑖 𝑘𝑘 𝑖𝑖 So𝐷𝐷 we𝑒𝑒⃗ collect𝐷𝐷�𝑒𝑒 ⃗results,𝜔𝜔 � 𝑒𝑒⃗ 𝑑𝑑𝜔𝜔 𝐷𝐷𝑒𝑒⃗ ∧ 𝜔𝜔 𝑒𝑒⃗ 𝑑𝑑𝜔𝜔 𝑒𝑒⃗ 𝜔𝜔 ∧ 𝜔𝜔 𝑒𝑒⃗ �𝑑𝑑𝜔𝜔 𝜔𝜔 ∧ 𝜔𝜔 �

= , = + (A.8) 2 𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑘𝑘 𝑖𝑖 𝑙𝑙 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑘𝑘 𝑖𝑖 where are the elements of the𝐷𝐷 two𝑒𝑒⃗ -form𝑒𝑒⃗ 𝜃𝜃 curvature𝜃𝜃 𝑑𝑑 𝜔𝜔. We 𝜔𝜔can ∧obtain𝜔𝜔 our formula Eq. A.6a for 𝑙𝑙 , 𝜃𝜃𝑖𝑖 Θ 𝑙𝑙 𝜃𝜃𝑖𝑖 = + = 𝑙𝑙 + (A.9) 𝑖𝑖𝑖𝑖 𝑙𝑙 𝑙𝑙 𝑗𝑗 𝑙𝑙 𝑚𝑚 𝑘𝑘 𝑗𝑗 𝜕𝜕Γ 𝑙𝑙 𝑘𝑘 𝑚𝑚 𝑗𝑗 𝑚𝑚 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖 𝜕𝜕𝑢𝑢 𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖 𝜃𝜃 𝑑𝑑�Γ 𝑑𝑑𝑢𝑢 � �Γ 𝑑𝑑𝑢𝑢 � ∧ �Γ 𝑑𝑑𝑢𝑢 � � Γ Γ � 𝑑𝑑𝑑𝑑 ∧ 𝑑𝑑𝑑𝑑 And we can extract the Riemann curvature tensor as above, being careful to explicitly anti- 𝑙𝑙 symmetrize in the indices m and j. 𝑅𝑅𝑖𝑖 𝑚𝑚𝑚𝑚

Recall from the discussion of parallel transport and holonomy in the textbook and Supplementary Lectures 9 and 11, that one can also obtain the Riemann curvature tensor from the commutator of components of the covariant derivative. Let’s sketch this calculation. We begin with Eq. A.4 for the covariant derivative of the components of a vector,

= = + 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝐷𝐷𝑣𝑣 𝜕𝜕𝑣𝑣 𝑖𝑖 𝑗𝑗 𝐷𝐷𝑘𝑘𝑣𝑣 𝑘𝑘 𝑘𝑘 Γ𝑗𝑗𝑗𝑗𝑣𝑣 Then a careful but straightforward calculation𝜕𝜕𝑢𝑢 produces,𝜕𝜕𝑢𝑢

[ , ] = (A.10) 𝑚𝑚 𝑚𝑚 𝑗𝑗 𝑘𝑘 𝑙𝑙 𝑗𝑗 𝑙𝑙𝑙𝑙 This formula was used extensively in the𝐷𝐷 textbook.𝐷𝐷 𝑣𝑣 Its𝑅𝑅 fundamental𝑣𝑣 feature is that the result of the commutator of the components of the covariant derivative applied to a vector is proportional to the vector itself. There are no derivatives of on the right-hand-side of Eq. A.10: the result 𝑚𝑚 is “ultra-local”. And the coefficient of the right𝑣𝑣-hand-side is the Riemann curvature tensor which is a property of the surface S, independent of . This equation was responsible for many of the 𝑚𝑚 fundamental properties of Riemann manifolds𝑣𝑣 presented in Supplementary Lectures 9 and 11. Next let’s derive the formula for the Christoffel symbols in terms of the metric on the surface S. From = + , we can extract the Christoffel symbols, 𝑘𝑘 𝑗𝑗 𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝜕𝜕 𝑒𝑒⃗ 𝑒𝑒⃗ Γ 𝑛𝑛�⃗𝑏𝑏 = , = 𝑘𝑘 𝑘𝑘 𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖 𝑙𝑙 𝑗𝑗 𝑖𝑖 𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 where = . These expressions𝑔𝑔 Γ suggest𝑒𝑒⃗ ∙ 𝜕𝜕 𝑒𝑒⃗ that 𝑔𝑔 canΓ be writtenΓ in terms of and its 𝑘𝑘 derivatives𝑔𝑔𝑘𝑘𝑘𝑘 𝑒𝑒⃗𝑘𝑘 ∙.𝑒𝑒 ⃗To𝑙𝑙 show this, begin with, Γ𝑖𝑖𝑖𝑖 𝑔𝑔𝑘𝑘𝑘𝑘 𝜕𝜕𝑖𝑖𝑔𝑔𝑘𝑘𝑘𝑘 31

= + = + (A.11)

𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑘𝑘 𝑗𝑗 𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗𝑗𝑗 Similarly, 𝜕𝜕 𝑔𝑔 �𝜕𝜕 𝑒𝑒⃗ ∙ 𝑒𝑒⃗ � �𝑒𝑒⃗ ∙ 𝜕𝜕 𝑒𝑒⃗ � Γ Γ

= + , = + (A.12)

𝑖𝑖 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑗𝑗 𝑘𝑘𝑘𝑘𝑘𝑘 𝑗𝑗 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖𝑖𝑖 To isolate we add the two𝜕𝜕 equations𝑔𝑔 Γ in Eq.Γ A.12 and𝜕𝜕 𝑔𝑔 subtractΓ Eq.Γ A.11. This gives, after some algebraΓ𝑖𝑖𝑖𝑖𝑖𝑖 which uses the symmetry = , Γ𝑖𝑖𝑖𝑖𝑖𝑖 Γ𝑗𝑗𝑗𝑗𝑗𝑗 = + (A.13a) 1 Γ𝑖𝑖𝑖𝑖𝑖𝑖 2 �𝜕𝜕𝑖𝑖𝑔𝑔𝑗𝑗𝑗𝑗 𝜕𝜕𝑗𝑗𝑔𝑔𝑘𝑘𝑘𝑘 − 𝜕𝜕𝑘𝑘𝑔𝑔𝑖𝑖𝑖𝑖� And raising the index k, we obtain a familiar result,

= + (A.13b) 𝑚𝑚 1 𝑚𝑚𝑚𝑚 Γ𝑖𝑖𝑖𝑖 2 𝑔𝑔 �𝜕𝜕𝑖𝑖𝑔𝑔𝑗𝑗𝑗𝑗 𝜕𝜕𝑗𝑗𝑔𝑔𝑘𝑘𝑘𝑘 − 𝜕𝜕𝑘𝑘𝑔𝑔𝑖𝑖𝑖𝑖� It is interesting in this context to consider how inner products vary around the surface. This exercise will lead to an important insight into the covariant derivative and the related subject of parallel transport which lies at the heart of curved manifolds and the Riemann curvature tensor, as we have seen in the textbook and in Supplementary Lectures 9 and 11. We calculate,

( ) = = + + 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 𝜕𝜕 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗( 𝜕𝜕)�=𝑔𝑔 𝑣𝑣 𝑤𝑤 +� �𝜕𝜕 𝑔𝑔 �+𝑣𝑣 𝑤𝑤 𝑔𝑔 �𝜕𝜕 𝑣𝑣+�𝑤𝑤 𝑔𝑔 𝑣𝑣 �𝜕𝜕 𝑤𝑤 � 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 ( ) =𝜕𝜕 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗ +�Γ Γ +�𝑣𝑣 𝑤𝑤 𝑔𝑔 +�𝜕𝜕 𝑣𝑣 �𝑤𝑤 =𝑔𝑔( 𝑣𝑣 �𝜕𝜕 𝑤𝑤) +� ( ) 𝑙𝑙 𝑖𝑖 𝑙𝑙 𝑗𝑗 𝑙𝑙 𝑗𝑗 𝑙𝑙 𝑖𝑖 𝑘𝑘 𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 𝑘𝑘 𝑘𝑘 𝑘𝑘 So, finally,𝜕𝜕 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗ 𝑔𝑔 �Γ 𝑣𝑣 𝜕𝜕 𝑣𝑣 �𝑤𝑤 𝑔𝑔 �Γ 𝑤𝑤 𝜕𝜕 𝑤𝑤 �𝑣𝑣 𝐷𝐷 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗ 𝑣𝑣⃗ ∙ 𝐷𝐷 𝑤𝑤��⃗

( ) = ( ) + ( ) (A.14)

𝑘𝑘 𝑘𝑘 𝑘𝑘 We learn the important fact that𝜕𝜕 the𝑣𝑣⃗ ∙differential𝑤𝑤��⃗ 𝐷𝐷 𝑣𝑣⃗ of∙ 𝑤𝑤 ��the⃗ inner𝑣𝑣⃗ ∙ product𝐷𝐷 𝑤𝑤��⃗ is determined by the covariant derivatives of the individual vectors,

( ) = = ( ) + ( ) (A.15) 𝑖𝑖 𝑗𝑗 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑘𝑘 So, if and 𝑑𝑑 𝑣𝑣⃗ ∙ vanish,𝑤𝑤��⃗ 𝜕𝜕 then�𝑔𝑔 𝑣𝑣 𝑤𝑤 is� 𝑑𝑑 consta𝑢𝑢 nt.𝐷𝐷 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗ 𝑑𝑑𝑢𝑢 𝑣𝑣⃗ ∙ 𝐷𝐷 𝑤𝑤��⃗ 𝑑𝑑𝑢𝑢

𝐷𝐷𝑘𝑘𝑣𝑣⃗ 𝐷𝐷𝑘𝑘𝑤𝑤��⃗ 𝑣𝑣⃗ ∙ 𝑤𝑤��⃗

We also recall from earlier discussions that if = 0 where = is the 𝑘𝑘 𝑘𝑘 𝑘𝑘 tangent to a curve ( ), then moves by parallel trans𝑝𝑝 𝐷𝐷port𝑘𝑘𝑣𝑣⃗ along the curve.𝑝𝑝 And𝑑𝑑𝑑𝑑 ⁄finally,𝑑𝑑𝑑𝑑 we 𝑘𝑘 discussed “metric compatibil𝑥𝑥 𝑠𝑠 ity”,𝑣𝑣⃗ = 0, in the textbook. We recognize that Eq. A.11 and A.12 can be written in just this form.𝐷𝐷𝑖𝑖𝑔𝑔 See𝑗𝑗𝑗𝑗 the textbook and Supplementary Lecture 11 for more detail.

Let’s end this appendix with a short discussion of curves, geodesics and the second fundamental form. Imagine a curve in R3 parametrized by its arc-length. Its path is given by ( ). Then the unit tangent to the curve is = . The rate at which turns determines the curve’s𝑥𝑥⃗ 𝑠𝑠 curvature , 𝑡𝑡⃗ 𝑑𝑑𝑥𝑥⃗⁄𝑑𝑑𝑑𝑑 𝑡𝑡⃗ 𝜅𝜅 = (A.16) 𝑑𝑑𝑡𝑡⃗ 𝑑𝑑𝑑𝑑 𝜅𝜅𝑁𝑁��⃗ where is a unit normal to . We studied curves in Supplementary Lectures 4 and 9, the intuition𝑁𝑁��⃗ behind Eq. A.16 and𝑡𝑡⃗ the geometric significance of .

Now consider a curve traveling on the surface S. In this𝜅𝜅 case it is useful to resolve into components in the tangent plane at and parallel to the normal to the surface there, 𝑑𝑑𝑡𝑡⃗⁄𝑑𝑑𝑑𝑑 𝑥𝑥⃗ 𝑛𝑛�⃗ = = + (A.17) 𝑑𝑑𝑡𝑡⃗ 𝑑𝑑𝑑𝑑 𝜅𝜅𝑁𝑁��⃗ 𝜅𝜅𝑔𝑔𝑘𝑘�⃗ 𝜅𝜅𝑛𝑛𝑛𝑛�⃗ where is a unit vector in the tangent plane and is the familiar unit normal to the surface. We ( ) can relate𝑘𝑘�⃗ and to the curve more explicitly𝑛𝑛�⃗ . Begin with, 𝜅𝜅𝑔𝑔 𝜅𝜅𝑛𝑛 𝑥𝑥⃗ 𝑠𝑠 = = = 𝑖𝑖 𝑖𝑖 𝑑𝑑𝑥𝑥⃗ 𝜕𝜕𝑥𝑥⃗ 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑡𝑡⃗ 𝑖𝑖 𝑒𝑒⃗𝑖𝑖 So, 𝑑𝑑𝑑𝑑 𝜕𝜕𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

= = = + = + + 𝑖𝑖 2 𝑖𝑖 𝑗𝑗 𝑖𝑖 2 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑑𝑑𝑡𝑡⃗ 𝑑𝑑 𝑑𝑑𝑢𝑢 𝑑𝑑 𝑢𝑢 𝜕𝜕𝑒𝑒⃗𝑖𝑖 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑑𝑑 𝑢𝑢 𝑙𝑙 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝜅𝜅𝑁𝑁��⃗ �𝑒𝑒⃗𝑖𝑖 � 𝑒𝑒⃗𝑖𝑖 2 𝑗𝑗 𝑒𝑒⃗𝑖𝑖 2 �𝑒𝑒⃗𝑙𝑙Γ𝑖𝑖𝑖𝑖 𝑛𝑛�⃗𝑏𝑏𝑖𝑖𝑖𝑖� Collecting, 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜕𝜕𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

+ = + + 2 𝑙𝑙 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑑𝑑 𝑢𝑢 𝑙𝑙 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝜅𝜅𝑔𝑔𝑘𝑘�⃗ 𝜅𝜅𝑛𝑛𝑛𝑛�⃗ 𝑒𝑒⃗𝑙𝑙 � 2 Γ𝑖𝑖𝑗𝑗 � 𝑛𝑛�⃗𝑏𝑏𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 33

So,

= + , = 2 𝑙𝑙 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 (A.18) 𝑑𝑑 𝑢𝑢 𝑙𝑙 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 2 𝜅𝜅𝑔𝑔𝑘𝑘�⃗ 𝑒𝑒⃗𝑙𝑙 � 𝑑𝑑𝑠𝑠 Γ𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝜅𝜅𝑛𝑛 𝑏𝑏𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 The first expression shows us that if the curve is a geodesic on S, = 0, then its differential equation is, 𝜅𝜅𝑔𝑔

+ = 0 2 𝑙𝑙 𝑖𝑖 𝑗𝑗 (A.19) 𝑑𝑑 𝑢𝑢 𝑙𝑙 𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 2 𝑑𝑑𝑑𝑑 Γ𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 This is a familiar result: since Eq. A.19 is a second order differential equation, it proves that there is a geodesic through every point on S pointing in any initial direction indicated by 𝑖𝑖 at = 0. In addition, the second relation in Eq. A.18 shows that is the ratio of the second𝑑𝑑𝑢𝑢 ⁄𝑑𝑑𝑑𝑑 fundamental𝑠𝑠 form to the first fundamental form, 𝜅𝜅𝑛𝑛

= 𝑖𝑖 𝑗𝑗 𝑏𝑏𝑖𝑖𝑖𝑖𝑑𝑑𝑢𝑢 𝑑𝑑𝑢𝑢 𝜅𝜅𝑛𝑛 2 another familiar result in classical differential geometry𝑑𝑑𝑑𝑑 which leads to some classic results on the Gaussian curvature K and the mean curvature H.

References

1. H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, 1963. 2. M. P. Do Carmo, Differential Forms and Applications, Springer-Verlag, Berlin, 1971. 3. M. Spivak, Calculus on Manifolds, Westview Press, Princeton, New Jersey, 1965.