Math 526 Differentiable Manifolds Akiva Weinberger Final project Riemannian geometry Abstract Riemannian geometry 1 is the “intrinsic” study of manifolds equipped with a metric , a way of measuring tangent vectors. From this relatively small amount of information we are able to recover a lot of geometric information about our manifold: in particular, we shall motivate and define the canonical affine connection of a Riemannian manifold and find its intrinsic curvature We will then use these tools to prove some simple theorems. Motivation Any map projection of the Earth will introduce distortions: there is no way to flatten a sphere onto a plane in such a way as to preserve lengths, angles, and areas. If we wish to use a map projection to calculate geometric quantities, we will need a way to quantify the distortion at every point. Similarly, the same is true for nearly any surface: projections will have distortions. (Two counterexamples are sections of cylinders and cones.) This raises a natural question: how much geometric information can we recover from the distortion data alone? Can we tell if the original shape was a sphere? If so, what was its radius? What is the shortest path between two points? Given a path, how much can we learn about its acceleration? First definitions A Riemannian metric on a differentiable manifold M is a correspondence which asso- ciates an inner product ⟨· , ·⟩ p of T p M to each point p ∈ M This metric is required to be differentiable, in the following sense: if ( U, ( x i )) is any smooth chart for M , then ⟨ ∂ ∂x i | p , ∂ ∂x j | p ⟩ p := g ij ( x 1 , . . . , x n ) is a differentiable function on φ − 1 ( U ). In coordinates, we may represent this as a matrix. If M is two-dimensional: g = [ g 11 g 12 g 21 g 22 ] Clearly, g is symmetric. Equivalently, the metric may be regarded as a symmetric covariant 2-tensor field on M that is positive-definite at each point. We may represent this this way, using a two- dimensional example for brevity: d s 2 = g 11 d x 2 + g 12 d x d y + g 21 d y d x + g 22 d y 2 = g 11 d x 2 +2 g 12 d x d y + g 22 d y 2 ( ds is to be interpreted as the length of a small change in position.) 1 Named after German mathematician Bernhard Riemann ( ["bEKnhaKt "Ki:man] ) (1826 – 1866). 1 Figure 1: (a) The Behrmann map projection with Tissot ellipses (b) Tissot ellipses of an arbitrary projection from the plane to itself We now wish to find a way to visualize a Riemannian metric. For a given chart φ : M → U ⊆ R n and a point p ∈ U , we may draw an infinitesimal circle around φ − 1 p and project it onto the chart. How will this look? If we identify T φ − 1 p M with R n , the set of vectors of length ε around p are those satisfying ⟨ v, v ⟩ p = v ⊤ gv = ε 2 (using the matrix representation of g ), which - since g is positive definite - is the equation of an ellipsoid (or an ellipse in two dimensions). These Tissot ellipsoids carry exactly the same information as the metric g The task of Riemannian geometry is now visually clarified. Instead of a 2-covector field, we may visualize an ellipsoid field (an association of ellipsoids to each point). Given such a field, we wish to determine the geometric properties of the surface or manifold it came from. Given Figure 1(b), it is not always easy to visually recognize a plane, so we get the feeling that this might not be a straightforward task. The Levi-Civita connection 2 To motivate what comes next, we will give our manifold some extrinsic geometric data, and see what constructions we can create that only depend on the intrinsic data. Suppose the manifold M has an isometric embedding i : M → R N , meaning one in which lengths are preserved. This allows us to identify T p M with a subspace of R N . This identification has a particularly simple form. Recall that if f is a function M → R , Xf is the directional derivative. This concept extends seamlessly to functions M → R N , and we can easily verify that X 7 → Xi is the identification we want. 3 Given a vector field Y on M and a vector X p , p ∈ M , we wish to consider the directional derivative of Y in the direction X p . In this notation, this is simply XY i p =: d Y d X | p . Is this an intrinsic concept? Unfortunately not: d Y d X | p may not even lie tangent to the manifold! For a simple example, consider the cylinder defined by x 2 + y 2 = 1 in R 2 . If Y is the ‘eastward’ vector field on a sphere Y ( x,y,z ) = ( − y, x, 0) and X (1 , 0 , 0) = (0 , 1 , 0), then d Y d X | p = ( − 1 , 0 , 0) points radially inward. 2 ["lE:vi "tSi:vita] Named after Italian mathematician Tullio Levi-Civita (1873 – 1941), rather than two separate people named Levi and Civita. 3 In the ensuing discussion, I will not be careful to distinguish between X and Xi , or in particular between ∇ X Y and ( ∇ X Y ) i 2 As a repair, we can consider the perpendicular projection onto the manifold. Using the notation ∥ to signify this projection, we define ∇ X Y = d Y d X | p ∥ = ( XY i ) ∥ (Assume for the time being that X is a vector field rather than a single vector.) We now hope that this is an intrinsic notion, that is, that it depends only upon g To see if ∇ is intrinsically defined, we list some properties. The following are all easy to verify: (i) ∇ f X + gY Z = f ∇ X Z + g ∇ Y Z (ii) ∇ X ( Y + Z ) = ∇ X Y + ∇ X Z (iii) ∇ X ( f Y ) = f ∇ X Y + Xf Y (iv) X ⟨ Y, Z ⟩ = ⟨∇ X Y, Z ⟩ + ⟨ Y, ∇ X Z ⟩ , where ⟨· , ·⟩ is given by the Riemannian metric on M The last one follows from the product rule d d X ⟨ Y, Z ⟩ = ⟨ dY dX , Z ⟩ + ⟨ Y, d Z d X ⟩ and ⟨ X, Y ∥ ⟩ = ⟨ X, Y ⟩ (recall that X is tangent to M ). There is one more property we will need: (v) ∇ X Y − ∇ Y X = [ X, Y ] Proof: ∇ X Y − ∇ Y X = ( XY i ) ∥ − ( Y Xi ) ∥ = (( XY − Y X ) i ) ∥ = [ X, Y ] ∥ i = [ X, Y ] i We are now ready to assemble the pieces. Recall that our goal is to show that ∇ is intrinsically defined. Consider the following three identities: X ⟨ Y, Z ⟩ = ⟨∇ X Y, Z ⟩ + ⟨ Y, ∇ X Z ⟩ (1) Y ⟨ Z, X ⟩ = ⟨∇ Y Z, X ⟩ + ⟨ Z, ∇ Y X ⟩ (2) Z ⟨ X, Y ⟩ = ⟨∇ Z X, Y ⟩ + ⟨ X, ∇ Z Y ⟩ (3) The equation (1) + (2) − (3), combined with several applications of (v), give us ⟨ Z, ∇ Y X ⟩ = 1 2 ( X ⟨ Y, Z ⟩ + Y ⟨ Z, X ⟩ − Z ⟨ X, Y ⟩ − ⟨ [ X, Z ] , Y ⟩ − ⟨ [ Y, Z ] , X ⟩ − ⟨ [ X, Y ] , Z ⟩ ) (4) This is known as the Koszul formula 4 Since a vector can be determined by its inner product with all other vectors (via, for example, an orthonormal basis), this proves that any ∇ satisfying (i)-(v) must be unique. In the other direction, given an arbitrary manifold, we may define ∇ by (4). Thus, ∇ is an intrinsic concept. A ∇ satisfying (i)-(iii) is called an (affine) connection . If it satisfies (iv), it is compatible with the metric If it satisfies (v) as well, it is torsion-free or symmetric Thus: on any 4 [kOsyl] . Named after French mathematician Jean-Louis Koszul (1921 – 2018). 3 Riemannian metric, there is a unique connection that is torsion-free and compatible with the metric . This connection is defined by (4) and is called the Levi-Civita connection (Note, by the way, that d Y d X also satisfies (4). It should make sense that, since ∇ was motivated by a projection to a subspace, the formula we derive should refer to inner products with that subspace.) We finish out this section with some coordinate work. Let X 1 := ∂ ∂x 1 , . . . , X n := ∂ ∂x n be coordinate vector fields, so that [ X i , X j ] = 0. Write ∇ X i X j = ∑ k Γ k ij X k (These are called Christoffel symbols 5 ) Let X = ∑ i x i X i and Y = ∑ j y j X j be arbitrary vector fields. Then some calculations involving (i)-(iii) tell us ∇ X Y = ∑ k ∑ ij x i y j Γ k ij + X ( y k ) X k (5) Note that to find ( ∇ X Y )( p ) we only need to know the value of X at p ; however, we need differential knowledge of Y From the Koszul formula (4) and [ X i , X j ] = 0, it follows that ∑ ℓ Γ ℓ ij g ℓk = 1 2 ( ∂ ∂x i g jk + ∂ ∂x j g ik − ∂ ∂x k g ij ) Write g ij for the entries of the inverse matrix of g . Then Γ m ij = 1 2 ∑ k ( ∂ ∂x i g jk + ∂ ∂x j g ik − ∂ ∂x k g ij ) g km Parallel transport We have learned how to directionally differentiate vector fields - at least, intrinsically. But why is this called a connection ? This suggests that we should somehow be able to “connect” two points on a manifold. Indeed, we can: given a curve c : I → M ( I some interval), we will find an isometry between T c ( t 0 ) M and T c ( t 1 ) Let V be a vector field along the curve c , that is, V ( t ) ∈ T c ( t ) M for t ∈ I Define D V d t = ∇ d c / d t V . (A priori this notation should not make sense, since neither d c d t nor V are vector fields on all of M , but (5) tells us that ∇ X Y only depends on the value of X at p and the value of Y on a curve through p . If M is embedded in R N , D V d t = ( d V d t ) ∥ .) A vector field V along a curve c : I → M is called parallel when D V d t = 0 for all t ∈ I Given a vector V 0 at c ( t 0 ), there exists a unique parallel vector field V along c such that V ( t 0 ) = V 0 V ( t ) is called the parallel transport of V ( t 0 ) along c . (We omit the proof, which is a straightforward application of the existence and uniqueness theorem for linear ordinary differential equations.) 5 Named for German mathematician Elwin Bruno Christoffel (1829 – 1900). 4 Suppose V and W are parallel along c . Then d d t ⟨ V, W ⟩ = 〈 D V d t , W 〉 + 〈 V, D W d t 〉 (by (iv)) = ⟨ 0 , W ⟩ + ⟨ V, 0 ⟩ = 0 Thus, ⟨ V, W ⟩ is constant along c . This justifies that parallel transport provides an isometry between T c ( t 0 ) M and T c ( t 1 ) M This provides a nice parallel between [ X, Y ] and ∇ X Y . Let θ be the flow of X , and let P t : T p M → T θ t ( p ) be the parallel transport along an integral curve of X . Then [ X, Y ] p = d d t (d θ − 1 t ( Y θ t ( p ) ) ( ∇ X Y ) p = d d t ( P − 1 t ( Y θ t ( p ) )) The former is familiar to us. Proof of the latter: Let E 1 , . . . , E n be an orthonormal basis of T p M , and extend it via parallel transport to a vector field along the integral curve. Then we may write Y θ t ( p ) = ∑ j y j E j . But then ∇ X Y = ∑ j ∇ X ( y j E j ) = ∑ j ( d y j d t E j + y j ∇ X E j ) = ∑ j d y j d t E j d d t ( P − 1 t ( Y θ t ( p ) )) = d d t P − 1 t ∑ j y j E j = d d t ∑ j y j E j = ∑ j d y j d t E j To help us in what follows, let’s find an analogue of equation (v), the symmetry of the connection, using D d t . A differentiable map s : A ⊆ R 2 → M is called a parametrized surface in M if A sufficiently nice 6 . A vector field V along s is a differentiable mapping associating to each q ∈ A a vector V ( q ) ∈ T s ( q ) M . If ( u, v ) are coordinates to R 2 , then we can write the coordinate vector fields of s as ∂s ∂u and ∂s ∂v . Then: D d u ∂s ∂u = D d v ∂s ∂v The proof is omitted; write everything in coordinates and use the symmetry of the connec- tion. In the case where M is embedded in R N , this is a direct application of the commuta- tivity of mixed partials: ( d d u ∂s ∂u ) ∥ = ( d d v ∂s ∂v ) ∥ Geodesics Geodesics are the “straightest curve possible” on a curved surface or manifold. On a sphere, these are the great circles. If M is an embedded manifold, they can be characterized by the 6 Connected, with U ⊆ A ⊆ U for some open set U , such that the boundary ∂A is a piecewise differentiable curve with whose interior vertex angles are not 0. 5 paths of particles which are not acted on by any force except those required to keep them on the manifold. This motivates the following definition. A curve γ is a geodesic if D d t ( d γ d t ) = 0. Since d d t ⟨ d γ d t , d γ d t ⟩ = 2 ⟨ D d t ( d γ d t ) , d γ d t ⟩ = 0, geodesics have constant speed. If this speed is 1, the geodesic is normalized One property of geodesics is that they locally minimize length. To see this, let’s employ the calculus of variations. The energy of a curve γ : I → M is E := ∫ t 1 t 0 | d γ d t | 2 d t Intuitively, imagine stretching a rubber band along γ ; this is its potential energy. This will be more convenient to work with than length. In fact, while all reparametrizations of a geodesic extremize length, only constant-speed geodesics extremize energy. The details are omitted, but an application of the Schwarz lemma tells us that L 2 = ( t 1 − t 0 ) E for constant speed curves and L 2 < ( t 1 − t 0 ) E otherwise; thus, to minimize length it is enough to minimize energy. Letting γ be a curve, a variation of γ is a parametrized surface s : ( − ε, ε ) × [ t 0 , t 1 ] → M such that s 0 ( t ) = γ ( t ) for all t (We will only consider differentiable variations; a more thorough treatment would incorporate piecewise differentiable variations as well.) The variation is proper if s u ( t 0 ) = γ ( t 0 ) and s u ( t 1 ) = γ ( t 1 ) for all u ∈ ( − ε, ε ). Then d E d u = d d u ∫ t 1 t 0 〈 d γ d t , d γ d t 〉 d t = 2 ∫ t 1 t 0 〈 D d u d γ d t , d γ d t 〉 d t = 2 ∫ t 1 t 0 〈 D d t d γ d u , d γ d t 〉 d t = − 2 ∫ t 1 t 0 〈 d γ d u , D d t d γ d t 〉 d t using integration by parts for that last equality. (Note the crucial use of the symmetry of the connection!) Since d γ d u is arbitrary, this means γ extremizes energy iff D d t d γ d t = 0, as desired. Curvature (motivation) Are all n -manifolds isometric, locally and to low order? Intuition suggests that spheres of different radii must not be isometric, not even locally. There must be some obstruction at each point. How many degrees of freedom does this obstruction have? Let us answer this question in two dimensions. How many degrees of freedom do we have available to choose a metric (a collection of Tissot ellipses) isometric to the plane? (Recall the crazy example in Figure 1(b).) The identity map is a chart of the plane around the origin. Locally, we may obtain any other arbitrary chart around the origin with any origin-preserving function f : R 2 → R 2 . Through the series expansion, f 1 ( x, y ) = a 1 x + a 2 y + a 11 x 2 + a 12 xy + a 22 y 2 + . . . f 2 ( x, y ) = b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 + . . . Thus, up to order two, f has ten degrees of freedom. (Since the choice of f (0 , 0) has no effect on the metric, it actually makes more sense to consider this to be “first-order”.) Since 6 precomposing with a rotation will not change the metric, up to first order we have nine degrees of freedom to choose a flat metric. On the other hand, consider an arbitrary Riemannian manifold M and a point p ∈ M Locally, we may choose g arbitrarily, as well as its first and second derivatives. Thus, the following are all free to vary: g 11 , g 12 , g 22 , ∂ ∂x g 11 , ∂ ∂x g 12 , ∂ ∂x g 22 , ∂ ∂y g 11 , ∂ ∂y g 12 , ∂ ∂y g 22 This is nine degrees of freedom. So up to first order, the plane has enough degrees of freedom to locally approximate any metric. Up to second order, things start to look different. Carefully counting, f has 17 degrees of freedom, while g has 18. We’ve lost a degree of freedom! This extra degree of freedom is the curvature . Only points with curvature 0 locally look like a plane up to second order. How may we quantify this curvature? To do this, let’s consider how fast geodesics spread apart. Jacobi fields 7 Given a point p and a vector v ∈ T p M , we may “follow” that vector, shooting out a geodesic γ with γ (0) = p and γ ′ (0) = v . Define exp p ( v ) to be γ (1). (This is equivalent to shooting out a normalized geodesic with γ ′ (0) = v/ | v | and asking for γ ( | v | ).) This is the exponential map , which encodes all geodesics coming from p 8 In T p M , let w be perpendicular to v , and imagine the vector field W = tw along the curve c ( t ) = tv (This is a vector field in the tangent space!) The motivation for this is that, much as a vector represents an infinitesimal displacement of a point, this vector field along the ray c represents an infinitesimal displacement (rotation, really) of the ray. Letting γ ( t ) = exp p ( tv ) be a geodesic, then, J ( t ) := ( d exp p ) tv ( tw ) represents an infinitesimal rotation of a geodesic . The size of J ( t ) then represents the rate at which nearby geodesics emitted from the same point spread apart. Letting s ( t, u ) = exp p ( tv + uw ), we see that ∂s ∂t are geodesics, ∂s ∂t ( t, 0) = γ ′ ( t ), and ∂s ∂u ( t, 0) = J ( t ). So 0 = D D u 0 = D D u D D t ∂s ∂t = D D t D D u ∂s ∂t + ( D D u D D t ∂s ∂t − D D t D D u ∂s ∂t ) = D D t D D t ∂s ∂u + ( D D u D D t ∂s ∂t − D D t D D u ∂s ∂t ) = D 2 J d t 2 + ( D D u D D t γ ′ − D D t D D u γ ′ ) This motivates us to consider the operation D D v D D u − D D u D D v very carefully. (It is not zero in general! In the embedded case, the projections mess with it, so we cannot apply commuta- tivity of partial derivatives.) 7 Carl Jacobi, (1804 – 1851) 8 This is a bit of an odd name in our context, especially since on M = R n , we have exp 0 = id. The name makes more sense in the context of Lie groups. In the meantime, think of it as a map T p M → M that’s trying to be “as close to the identity map as possible”. 7 Curvature (again) Define R ( X, Y ) Z = ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [ X,Y ] Z. Sine [ ∂s ∂u , ∂s ∂t ] = 0, we see that the Jacobi equation above can be written as D 2 J d t 2 + R ( γ ′ , J ) γ ′ = 0 But why the ∇ [ X,Y ] Z term on the end? It turns out that ∇ Y ∇ X Z − ∇ X ∇ Y Z is not fully linear in its arguments, but ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [ X,Y ] Z is . Thus, if we wrote R in coordinates, it does not depend on any derivatives of the coordinates x i , y i , or z i . The result is that R ( X, Y ) Z can take vectors as inputs rather than vector fields; R is a (3 , 1)-tensor. I won’t justify it here, but R has a very nice geometric interpretation. Draw a small rectangle in M by flowing forward a bit along X , flowing forward a bit along Y , flowing backwards a bit along X , and flowing backwards a bit along Y . This doesn’t actually close up in general (not even to first order); we must flow along [ X, Y ] to draw a closed loop. If we parallel transport Z along this loop, up to first order it ends up as the same as it started, but up to second order it will have rotated slightly. (This rotation is known as holonomy .) This small change is R ( X, Y ) Z If M is two-dimensional, we can extract a single quantity out of this. Consider two orthonormal vectors X, Y ∈ T p M and parallel transport X along a small loop in the same manner. Since it will come back slightly rotated, we can extract the amount of rotation by taking the inner product with Y We thus define the curvature of M to be K := ⟨ R ( X, Y ) X, Y ⟩ . (If X and Y are not orthonormal, this will be off by a factor of the square of the area of the parallelogram they span.) Returning to the Jacobi equation, suppose K is constant and that γ is normalized. Since w was perpendicular to v , a calculation reveals that J is perpendicular to γ ′ . Thus: D 2 J d t 2 + KJ = 0 The curvature K acts as a “force” compressing J ! This should make sense intuitively: on a plane, infinitesimally adjacent geodesics should grow apart linearly, but on a sphere, they should reunite every half trip around the sphere and so | J | should oscillate. In a negatively curved surface, | J | grows approximately exponentially. Omitted topics There were many things I wanted to get to but could not due to time constraints. One was to motivate the formula for R better. I also wanted to discuss sectional curvature in dimension higher than two. For example, in dimensions 3 and above, curvature cannot be described by a single number, though we can consider the so-called “sectional curvature” along dimension-2 subspaces of T p M , and we may take their average to obtain a quantity known as “Ricci curvature”. Another was to prove Hadamard’s theorem: if M has nonpositive sectional curvature everywhere, then its universal cover is diffeomorphic to R n , and the exponential map pro- vides the diffeomorphism. The proof sketch is to use the Jacobi equation to show that | J | 8 increases, and to show that failures of exp to be a local diffeomorphism correspond exactly to solutions to J ( t ) = 0, t ̸ = 0. A third was to prove the Bonnet–Myers theorem: if the Ricci curvature is bounded below by 1 /r 2 for some constant r , then the diameter of the manifold is at most πr , a bound obtained by the sphere. 9 The proof involves computing d 2 E d u 2 for some variation of a geodesic ( E is the energy), rewriting it to involve R in a similar manner to the derivation of the Jacobi equation, and determining that d 2 E d u 2 must be negative if the geodesic is too long. References [1] do Carmo, Manfredo Perdig ̃ ao (1992). Riemannian geometry. Mathematics: The- ory & Applications. Translated from the second Portuguese edition by Francis Fla- herty. Boston, MA: Birkh ̈ auser Boston, Inc. ISBN 0-8176-3490-8. MR 1138207. Zbl 0752.53001. 9 Amusingly, the “diameter” of the unit sphere is considered to be π in Riemannian geometry rather than 2. This is because we measure distances along the sphere rather than through 9