Riemannian geometry of surfaces In this note, we will learn how to make sense of the concepts of differential geometry on a surface M, which is not necessarily situated in R 3. This intrinsic approach is referred to as Riemannian Geometry. In a nutshell, the idea is to turn every theorem about surfaces in R 3 into a definition. Doing so, will guarantee that all our results from before will still be valid in the more general context. Definition. [Abstract surfaces] Let M be a set and P a collection of one-to-one functions x : D M from open subsets D of R 2 into M. We call M an abstract surface and the functions of P the patches on M, if the following properties are satisfied: (i) [Covering axiom] For every p M there is some x : D M of the collection P such that p x(d); (ii) [Smooth overlap axiom] For every two functions x : D M and y : E M of P, the composite functions x 1 y and y 1 x have domains which are open subsets of R 2 and are differentiable on these domains; (iii) [Maximality axiom] If x : D M is any one-to-one function, which when added to the collection P makes the new collection satisfy Properties (i) and (ii), then x : D M already belongs to P; (iv) [Hausdorff axiom] If p, q M with p q, then there are functions x : D M and y : E M of P such that p x(d), q y(e) and x(d) y(e) =. Definition. [Open sets] Let M be an abstract surface. A subset S of M is called open relative to M if for every p S there is a patch x : D M such that p x(d) S. Remark. We usually assume that an abstract surface M can be covered by countably many images of patches. This ensures that M can be turned into a metric space. Definition. [Differentiability] (i) A function F : R M from an open subset R of some Euclidean space R n into an abstract surface M is called differentiable if for every p R and for every patch x : D M with F (p) x(d), there is an open subset S R such that p S, F (S) x(d), and x 1 F : S R 2 is differentiable. (ii) A function F : M R n from an abstract surface into some Euclidean space is called differentiable if for every p M and every patch x : D M with p x(d) the function F x : D R n is differentiable. (iii) A function F : M N from an abstract surface M to an abstract surface N is called differentiable if for every p M and every patch y : E N with F (p) y(e) there is a patch x : D M such that p x(d), F (x(d)) y(e), and y 1 F x : D R 2 is differentiable.
Note. Recall that differentiability needs to be checked on only one patch at a time, because of the smooth overlap axiom. Definition. [Curves] A differentiable function α : I M from an open interval I into an abstract surface M is called a curve. The differentiable functions (a 1, a 2 ) = x 1 α : I D are called the coordinate functions of α with respect to the patch x : D M. The coordinate functions of α are the unique functions with α(t) = x(a 1 (t), a 2 (t)). Notation. As before, we will denote the collection of differentiable real-valued functions f : M R by F(M). Definition. [Tanget vectors] Let α : I M be a curve in an abstract surface M. For each t I, the velocity vector α (t) is defined to be the function α (t) : F(M) R given by α (t)[f] = (f α) (t). A tangent vector v p at p M is the velocity vector α (0) of some curve α : I M with α(0) = p. The collection T p (M) = {α (0) α : I M is a curve with α(0) = p} is called the tangent plane to M at p. Definition. [Basis for the tangent plane] Let x : D M be a patch on an abstract surface M and x(u 0, v 0 ) = p M. Consider the parameter curves α : I M and β : I M, which are given by α(t) = x(t, v 0 ) and β(t) = x(u 0, t), respectively. Then the velocity vectors α (u 0 ) and β (v 0 ) are denoted by x u (u 0, v 0 ) and x v (u 0, v 0 ), respectively. By the remark below, {x u (u 0, v 0 ), x v (u 0, v 0 )} is a basis for the tangent plane T p (M). Remark. Observe that x u (u 0, v 0 )[f] = α (u 0 )[f] = (f α) (u 0 ) = We therefore adapt the usual abbreviation of (f x) (u 0, v 0 ). u x u [f] = f u and x v[f] = f v. (1) Given any curve γ : I M, we can write γ in terms of its coordinate functions with respect to the patch x : D M, say γ(t) = x(a 1 (t), a 2 (t)). Then we get γ (t) = a 1(t)x u (a 1 (t), a 2 (t)) + a 2(t)x v (a 1 (t), a 2 (t)). To check this, simply verify that both sides yield the same real number when applied to a real-valued differentiable function, that is, γ (t)[f] = a 1(t)x u (a 1 (t), a 2 (t))[f] + a 2(t)x v (a 1 (t), a 2 (t))[f] for all f F(M). In turn, this follows from Equation (1) and the usual chain rule from Calculus. Therefore, one shows, exactly as before, that x u (u 0, v 0 ) and x v (u 0, v 0 ) form a basis for T p (M). If two patches x : D M and y : E M overlap in the set G = x(d) y(e) and p G, then one shows, as before, that the matrix which converts the coordinates in T p (G) from the x-patch basis to the y-patch basis, is given by the derivative of y 1 x, precomposed with x 1.
Definition. [Tangent vector fields] A tangent vector field on an abstract surface M is a function V, which assigns to each p M a tangent vector V (p) T p (M). If x : D M is a patch, then there are unique functions v 1, v 2 : x(d) R with V (p) = v 1 (p)x u x 1 (p) + v 2 (p)x v x 1 (p). We only consider tangent vector fields V for which v 1 and v 2 are differentiable. we denote the collection of all tangent vector fields on M by V(M). Definition. [Differential forms] Differential forms, wedge products, exterior derivatives, and pullbacks are defined exactly as before. The next definition replaces the dot product by the notion of a Riemannian metric: Definition. [Riemannian metric] Given an abstract surface M, a Riemannian metric is a function which assigns at every point p M to every pair v p, w p T p (M) a real number v p, w p p such that for all u p, v p, w p T p (M) and all real numbers c 1 and c 2 (i) c 1 u p + c 2 v p, w p p = c 1 u p, w p p + c 2 v p, w p p and u p, c 1 v p + c 2 w p p = c 1 u p, v p p + c 2 u p, w p p ; (ii) v p, w p p = w p, v p p ; (iii) v p, v p p 0; (iv) v p, v p p = 0 if and only if v p = 0 p. We also require that for every pair of tangent vector fields V, W V(M), the realvalued function f : M R given by f(p) = V (p), W (p) p is differentiable. In the future, we will simply write v p, w p for v p, w p p. Definition. [Norm, length, and angle] If M is an abstract surface with Riemannian metric,, then the norm of a tangent vector v p T p (M) is defined by v p = v p, v p. Using this norm, the length of a curve segment α : [a, b] M is defined to be b a α (t) dt. The angle between two non-zero tangent vectors v p, w p T p (M) is defined to be the unique θ [0, π] with cos θ = v p, w p v p w p. Remark. In Linear Algebra one shows that the (inner product) function, satisfies the so-called Cauchy-Schwarz Inequality: v p, w p v p w p. Moreover, it is shown how this implies the Triangle Inequality: v p + w p v p + w p.
The most common way of putting a Riemannian metric on an abstract surface is the following: Exercise. Let x : D M be a patch of an abstract surface M and suppose that E, F, G : D R are any differentiable functions such that E > 0, G > 0, and EG F 2 > 0. Then there is exactly one Riemannian metric, on x(d) with the property that E = x u, x u, F = x u, x v, and G = x v, x v. Definition. [Frame fields and dual forms] A pair E 1, E 2 of tangent vector fields on an abstract surface M with Riemannian metric, is called a frame field, if E 1, E 1 = 1, E 1, E 2 = 0, and E 2, E 2 = 1. The dual forms of a frame field E 1, E 2 are defined by θ 1 (v p ) = v p, E 1 (p) and θ 2 (v p ) = v p, E 2 (p). Definition. [Connection form] A frame field E 1, E 2 on an abstract surface M with Riemannian metric, with dual forms θ 1 and θ 2, has only one connection form, which is defined by the formula ω 12 (V ) = v 1 dθ 1 (E 1, E 2 ) + v 2 dθ 2 (E 1, E 2 ) for a tangent vector field V = v 1 E 1 + v 2 E 2. We also put ω 21 = ω 12. Remark. As we proved in class, the formula for ω 12 is the only 1-form on M that satisfies dθ 1 = ω 12 θ 2 and dθ 2 = ω 21 θ 1. Definition. [Gaussian curvature] Let E 1, E 2 be a tangent frame field on an abstract (Riemannian) surface M with dual forms θ 1 and θ 2 and connection form ω 12. Since every 2-form η on M can be written (uniquely) in the form η = fθ 1 θ 2 with f = η(e 1, E 2 ), we may define the Gaussian curvature function on M by the equation dω 12 = Kθ 1 θ 2. Remark. One has to verify that the definition of K does not depend on the choice of the frame field E 1, E 2. This is done in Lemma 1.4 and Theorem 2.1 of Chapter 7. Definition. [Covariant derivative] Let E 1, E 2 be a frame field on an abstract (Riemannian) surface M with dual forms θ 1 and θ 2, and connection form ω 12. For tangent vector fields V, W V(M), say W = f 1 E 1 + f 2 E 2, we define the covariant derivative by V W = (V [f 1 ] + f 2 ω 21 (V ))E 1 + (V [f 2 ] + f 1 ω 12 (V ))E 2. Remark. Recall that this is the (2-dimensional analog of the) covariant derivative formula of #5 2.7. Again, one has to verify that this definition does not depend on the choice of frame field. This is done in Theorem 3.2 of Chapter 7. Indeed, there you will find a proof of the following Theorem.
Theorem. For every abstract surface M with Riemannian metric,, there exists one and only one covariant derivative that has the following properties: (i) fv +gw Y = f V Y + g W Y for every f, g F(M) and V, W, Y V(M); (ii) V (Y + Z) = V Y + V Z for every V, Y, Z V(M); (iii) V (fy ) = V [f]y + f V Y for every f F(M) and V, Y V(M); (iv) V [ Y, Z ] = V Y, Z + Y, V Z for every V, Y, Z V(M); (v) ω 12 (V ) = V E 1, E 2 for every frame field E 1, E 2, its ω 12, and every V V(M), and this is exactly the covariant derivative defined above, locally by any frame field. Remark. In Lemma 3.8 of Chapter 7 it is shown that the above intrinsic definition of covariant derivative coincides with that component of the regular 3-space covariant derivative, which is tangent to the surface, in the event that M is situated in R 3. In other words, it is exactly the derivative D V W from Problem 7 of Exam II.