Basic Riemannian Geometry

Basic Riemannian Geometry F.E. Burstall Department of Mathematical Sciences University of Bath Introduction My mission was to describe the basics of Riemannian geometry in just three hours of lectures, starting from scratch. The lectures were to provide back- ground for the analytic matters covered elsewhere during the conference and, in particular, to underpin the more detailed (and much more professional) lectures of Isaac Chavel. My strategy was to get to the point where I could state and prove a Real Live Theorem: the Bishop Volume Comparison The- orem and Gromov’s improvement thereof and, by appalling abuse of OHP technology, I managed this task in the time alloted. In writing up my notes for this volume, I have tried to retain the breathless quality of the original lectures while correcting the mistakes and excising the out-right lies. I have given very few references to the literature in these notes so a few remarks on sources is appropriate here. The first part of the notes deals with analysis on differentiable manifolds. The two canonical texts here are Spivak [5] and Warner [6] and I have leaned on Warner’s book in particular. For Riemannian geometry, I have stolen shamelessly from the excellent books of Chavel [1] and Gallot–Hulin–Lafontaine [3]. In particular, the proof given here of Bishop’s theorem is one of those provided in [3]. 1 What is a manifold? What ingredients do we need to do Differential Calculus? Consider first the notion of a continuous function: during the long process of abstraction and generalisation that leads from Real Analysis through Metric Spaces to Topology, we learn that continuity of a function requires no more structure on the domain and co-domain than the idea of an open set. By contrast, the notion of differentiability requires much more: to talk about the difference quotients whose limits are partial derivatives, we seem to require that the (co-)domain have a linear (or, at least, affine) structure. 1 However, a moment’s thought reveals that differentiability is a completely local matter so that all that is really required is that the domain and co- domain be locally linear, that is, each point has a neighbourhood which is homeomorphic to an open subset of some linear space. These ideas lead us to the notion of a manifold : a topological space which is locally Euclidean and on which there is a well-defined differential calculus. We begin by setting out the basic theory of these spaces and how to do Analysis on them. 1.1 Manifolds Let M be a Hausdorff, second countable1, connected topological space. M is a Cr manifold of dimension n if there is an open cover {Uα}α∈I of M and homeomorphisms xα : Uα → xα(Uα) onto open subsets of Rn such that, whenever Uα ∩ Uβ 6= ∅, xα ◦ x−1β : xβ(Uα ∩ Uβ)→ xα(Uα ∩ Uβ) is a Cr diffeomorphism. Each pair (Uα, xα) called a chart. Write xα = (x1, . . . , xn). The xi : Uα → R are coordinates. 1.1.1 Examples 1. Any open subset U ⊂ Rn is a C∞ manifold with a single chart (U, 1U ). 2. Contemplate the unit sphere Sn = {v ∈ Rn+1 : ‖v‖ = 1} in Rn+1. Orthogonal projection provides a homeomorphism of any open hemi- sphere onto the open unit ball in some hyperplane Rn ⊂ Rn+1. The sphere is covered by the (2n + 2) hemispheres lying on either side of the coordinate hyperplanes and in this way becomes a C∞ manifold (exercise!). 3. A good supply of manifolds is provided by the following version of the Implicit Function Theorem [6]: Theorem. Let f : Ω ⊂ Rn → R be a Cr function (r ≥ 1) and c ∈ R a regular value, that is, ∇f(x) 6= 0, for all x ∈ f−1{c}. Then f−1{c} is a Cr manifold. Exercise. Apply this to f(x) = ‖x‖2 to get a less tedious proof that Sn is a manifold. 1This means that there is a countable base for the topology of M . 2 4. An open subset of a manifold is a manifold in its own right with charts (Uα ∩ U, xα|Uα∩U ). 1.1.2 Functions and maps A continuous function f : M → R is Cr if each f ◦ x−1α : xα(Uα) → R is a Cr function of the open set xα(Uα) ⊂ Rn. We denote the vector space of all such functions by Cr(M). Example. Any coordinate function xi : Uα → R is Cr on Uα. Exercise. The restriction of any Cr function on Rn+1 to the sphere Sn is Cr on Sn. In the same way, a continuous map φ : M → N of Cr manifolds is Cr if, for all charts (U, x), (V, y) of M and N respectively, y ◦ φ ◦ x−1 is Cr on its domain of definition. A slicker formulation2 is that h ◦ φ ∈ Cr(M), for all h ∈ Cr(M). At this point, having made all the definitions, we shall stop pretending to be anything other than Differential Geometers and henceforth take r =∞. 1.2 Tangent vectors and derivatives We now know what functions on a manifold are and it is our task to dif- ferentiate them. This requires some less than intuitive definitions so let us step back and remind ourselves of what differentiation involves. Let f : Ω ⊂ Rn → R and contemplate the derivative of f at some x ∈ Ω. This is a linear map dfx : Rn → R. However, it is better for us to take a dual point of view and think of v ∈ Rn is a linear map v : C∞(M)→ R by vf def= dfx(v). The Leibniz rule gives us v(fg) = f(x)v(g) + v(f)g(x). (1.1) Fact. Any linear v : C∞(Ω)→ R satisfying (1.1) arises this way. Now let M be a manifold. The preceding analysis may give some motivation to the following 2It requires a little machinery, in the shape of bump functions, to see that this is an equivalent formulation. 3 Definition. A tangent vector at m ∈ M is a linear map ξ : C∞(M) → R such that ξ(fg) = f(m)ξ(g) + ξ(f)g(m) for all f, g ∈ C∞(M). Denote by Mm the vector space of all tangent vectors at m. Here are some examples 1. For γ : I →M a (smooth) path with γ(t) = m, define γ′(t) ∈Mm by γ′(t)f = (f ◦ γ)′(t). Fact. All ξ ∈Mm are of the form γ′(t) for some path γ. 2. Let (U, x) be a chart with coordinates x1, . . . , xn and x(m) = p ∈ Rn. Define ∂i|m ∈Mm by ∂i|mf = ∂(f ◦ x−1) ∂xi ∣∣∣∣ p Fact. ∂1|m, . . . , ∂n|m is a basis for Mm. 3. For p ∈ U ⊂ Rn open, we know that Up is canonically isomorphic to Rn via vf = dfp(v) for v ∈ Rn. 4. Let M = f−1{c} be a regular level set of f : Ω ⊂ Rn → R. One can show that Mm is a linear subspace of Ωm ∼= Rn. Indeed, under this identification, Mm = {v ∈ Rn : v ⊥ ∇fm}. Now that we have got our hands on tangent vectors, the definition of the derivative of a function as a linear map on tangent vectors is almost tauto- logical: Definition. For f ∈ C∞(M), the derivative dfm : Mm → R of f at m ∈M is defined by dfm(ξ) = ξf. 4 We note: 1. Each dfm is a linear map and the Leibniz Rule holds: d(fg)m = g(m)dfm + f(m)dgm. 2. By construction, this definition coincides with the usual one when M is an open subset of Rn. Exercise. If f is a constant map on a manifold M , show that each dfm = 0. The same circle of ideas enable us to differentiate maps between manifolds: Definition. For φ : M → N a smooth map of manifolds, the tangent map dφm : Mm → Nφ(m) at m ∈M is the linear map defined by dφm(ξ)f = ξ(f ◦ φ), for ξ ∈Mm and f ∈ C∞(N). Exercise. Prove the chain rule: for φ : M → N and ψ : N → Z and m ∈M , d(ψ ◦ φ)m = dψφ(m) ◦ dφm. Exercise. View R as a manifold (with a single chart!) and let f : M → R. We now have two competing definitions of dfm. Show that they coincide. The tangent bundle of M is the disjoint union of the tangent spaces: TM = ∐ m∈M Mm. 1.3 Vector fields Definition. A vector field is a linear map X : C∞(M) → C∞(M) such that X(fg) = f(Xg) + g(Xf). Let Γ(TM) denote the vector space of all vector fields on M . We can view a vector field as a map X : M → TM with X(m) ∈ Mm: indeed, we have X|m ∈Mp 5 where X|mf = (Xf)(m). In fact, vector fields can be shown to be exactly those maps X : M → TM with X(m) ∈ Mm which satisfy the additional smoothness constraint that for each f ∈ C∞(M), the function m 7→ X(m)f is also C∞. The Lie bracket of X,Y ∈ Γ(TM) is [X,Y ] : C∞(M)→ C∞(M) given by [X,Y ]f = X(Y f)− Y (Xf). The point of this definition is contained in the following Exercise. Show that [X,Y ] ∈ Γ(TM) also. The Lie bracket is interesting for several reasons. Firstly it equips Γ(TM) with the structure of a Lie algebra; secondly, it, and operators derived from it, are the only differential operators that can be defined on an arbitrary manifold without imposing additional structures such as special coordinates, a Riemannian metric, a complex structure or a symplectic form. There is an extension of the notion of vector field that we shall need later on: Definition. Let φ : M → N be a map. A vector field along φ is a map X : M → TN with X(m) ∈ Nφ(m), for all m ∈ M , which additionally satisfies a smoothness assumption that we shall gloss over. Denote by Γ(φ−1TN) the vector space of all vector fields along φ. Here are some examples: 1. If c : I → N is a smooth path then c′ ∈ Γ(φ−1TN). 2. More generally, for φ : M → N and X ∈ Γ(TM), dφ(X) ∈ Γ(φ−1TN). Here, of course, dφ(X)(m) = dφm(X|m). 3. For Y ∈ Γ(TN), Y ◦ φ ∈ Γ(φ−1TN). 6 1.4 Connections We would like to differentiate vector fields but as they take values in differ- ent vector spaces at different points, it is not so clear how to make difference quotients and so derivatives. What is needed is some extra structure: a con- nection which should be thought of as a “directional derivative” for vector fields. Definition. A connection on TM is a bilinear map TM × Γ(TM)→ TM (ξ,X) 7→ ∇ξX such that, for ξ ∈Mm, X,Y ∈ Γ(TM) and f ∈ C∞(M), 1. ∇ξX ∈Mm; 2. ∇ξ(fX) = (ξf)X|m + f(m)∇ξX; 3. ∇XY ∈ Γ(TM). A connection on TM comes with some additional baggage in the shape of two multilinear maps: Tm : Mm ×Mm →Mm Rm : Mm ×Mm ×Mm →Mm given by Tm(ξ, η) = ∇ξY −∇ηX − [X,Y ]|m Rm(ξ, η)ζ = ∇η∇XZ −∇ξ∇Y Z −∇[Y,X]|m where X, Y , Z ∈ Γ(TM) with X|m = ξ, Y|m = η and Z|m = ζ. Tm and Rm are, respectively, the torsion and curvature at m of ∇. Fact. R and T are well-defined—they do not depend of the choice of vector fields X, Y and Z extending ξ, η and ζ. We have some trivial identities: T (ξ, η) = −T (η, ξ) R(ξ, η)ζ = −R(η, ξ)ζ. and, if each Tm = 0, we have the less trivial First Bianchi Identity : R(ξ, η)ζ +R(ζ, ξ)η +R(η, ζ)ξ = 0. 7 A connection ∇ on TN induces a similar operator on vector fields along a map φ : M → N . To be precise, there is a unique bilinear map TM × Γ(φ−1TN)→ TN (ξ,X) 7→ φ−1∇ξX such that, for ξ ∈Mm, X ∈ Γ(TM), Y ∈ Γ(φ−1TN) and f ∈ C∞(M), 1. φ−1∇ξY ∈ Nφ(m); 2. φ−1∇ξ(fY ) = (ξf)Y|φ(m) + f(m)φ−1∇ξY ; 3. φ−1∇XY ∈ Γ(φ−1TN) (this is a smoothness assertion); 4. If Z ∈ Γ(TN) then Z ◦ φ ∈ Γ(φ−1TN) and φ−1∇ξ(Z ◦ φ) = ∇dφm(ξ)Z. φ−1∇ is the pull-back of ∇ by φ. The first three properties just say that φ−1∇ behaves like ∇, it is the last that essentially defines it in a unique way. 2 Analysis on Riemannian manifolds 2.1 Riemannian manifolds A rich and useful geometry arises if we equip each Mm with an inner product: Definition. A Riemannian metric g on M is an inner product gm on each Mm such that, for all vector fields X and Y , the function m 7→ gm(X|m, Y|m) is smooth. A Riemannian manifold is a pair (M, g) with M a manifold and g a metric on M . Here are some (canonical) examples: 1. Let ( , ) denote the inner product on Rn. An open U ⊂ Rn gets a Riemannian metric via Um ∼= Rn: gm(v, w) = (v, w). 8 2. Let Sn ⊂ Rn+1 be the unit sphere. Then Snm ∼= m⊥ ⊂ Rn+1 and so gets a metric from the inner product on Rn+1. 3. Let Dn ⊂ Rn be the open unit disc but define a metric by gz(v, w) = 4(v, w) (1− |z|2)2 (Dn, g) is hyperbolic space. Much of the power of Riemannian geometry comes from the fact that there is a canonical choice of connection. Consider the following two desirable properties for a connection ∇ on (M, g): 1. ∇ is metric: Xg(Y, Z) = g(∇XY,Z) + g(Y,∇XZ). 2. ∇ is torsion-free: ∇XY −∇YX = [X,Y ] Theorem. There is a unique torsion-free metric connection on any Rie- mannian manifold. Proof. Assume that g is metric and torsion-free. Then g(∇XY,Z) = Xg(Y, Z)− g(Y,∇XZ) = Xg(Y, Z)− g(Y, [X,Z])− g(Y,∇ZX) . . . and eventually we get 2g(∇XY,Z) = Xg(Y, Z) + Y g(Z, Y )− Zg(X,Y ) − g(X, [Y,Z]) + g(Y, [Z,X]) + g(Z, [X,Y ]). (2.1) This formula shows uniqueness and, moreover, defines the desired connec- tion. This connection is the Levi–Civita connection of (M, g). For detailed computations, it is sometimes necessary to express the metric and Levi–Civita connection in terms of local coordinates. So let (U, x) be a chart and ∂1, . . . , ∂n be the corresponding vector fields on U . We now define gij ∈ C∞(U) by gij = g(∂i, ∂j) and Christoffel symbols Γkij ∈ C∞(U) by ∇∂i∂j = ∑ k Γkij∂k. 9 (Recall that ∂1|m, . . . , ∂n|m form a basis for Mm.) Now let (gij) be the matrix inverse to (gij). Then the formula (2.1) for ∇ reads: Γkij = 1 2 ∑ l gkl(∂igjl + ∂jgli − ∂lgij) (2.2) since the bracket terms [∂i, ∂j ] vanish (exercise!). 2.2 Differential operators The metric and Levi–Civita connection of a Riemannian manifold are pre- cisely the ingredients one needs to generalise the familiar operators of vector calculus: The gradient of f ∈ C∞(M) is the vector field grad f such that, for Y ∈ Γ(TM), g(grad f, Y ) = Y f. Similarly, the divergence of X ∈ Γ(TM) is the function div f ∈ C∞(M) defined by: (div f)(m) = trace(ξ → ∇ξX) Finally, we put these together to introduce the hero of this volume: the Laplacian of f ∈ C∞(M) is the function ∆f = div grad f. In a chart (U, x), set g = det(gij). Then grad f = ∑ i,j gij(∂if)∂j and, for X = ∑ iXi∂i, divX = ∑ i ( ∂iXi + ∑ j ΓiijXj ) = 1√ g ∑ j ∂j( √ gXj). Here we have used ∑ i Γ i ij = (∂j √ g)/ √ g which the Reader is invited to deduce from (2.2) together with the well-known formula for a matrix-valued function A: d ln detA = traceA−1dA. 10 In particular, we conclude that ∆f = 1√ g ∑ i,j ∂i( √ ggij∂jf) = ∑ i,j gij(∂i∂jf − Γkij∂kf). 2.3 Integration on Riemannian manifolds 2.3.1 Riemannian measure (M, g) has a canonical measure dV on its Borel sets which we define in steps: First let (U, x) be a chart and f : U → R a measureable function. We set∫ U f dV = ∫ x(U) (f ◦ x−1) √ g ◦ x−1 dx1 . . .dxn. Fact. The change of variables formula ensures that this integral is well- defined on the intersection of any two charts. To get a globally defined measure, we patch things together with a partition of unity : since M is second countable and locally compact, it follows that every open cover of M has a locally finite refinement. A partition of unity for a locally finite open cover {Uα} is a family of functions φα ∈ C∞(M) such that 1. supp(φα) ⊂ Uα; 2. ∑ α φα = 1. Theorem. [6, Theorem 1.11] Any locally finite cover has a partition of unity. Armed with this, we choose a locally finite cover of M by charts {(Uα, xα)}, a partition of unity {φα} for {Uα} and, for measurable f : M → R, set∫ M f dV = ∑ α ∫ Uα φαf dV. Fact. This definition is independent of all choices. 11 2.3.2 The Divergence Theorem Let X ∈ Γ(TM) have support in a chart (U, x).∫ M divX dV = ∫ U 1√ g ∂i( √ gXi) dV = ∫ x(U) (∂i √ gXi) ◦ x−1 dx1 . . .dxn = ∫ x(U) ∂ ∂xi ( √ gXi) ◦ x−1 dx1 . . .dxn = 0. A partition of unity argument immediately gives: Divergence Theorem I. Any compactly supported vector field X on M has ∫ M divX dV = 0. Just as in vector calculus, the divergence theorem quickly leads to Green’s formulae. Indeed, for f, h ∈ C∞(M), X ∈ Γ(TM) one easily verifies: div(fX) = f divX + g(grad f,X) whence div(f gradh) = f∆h+ g(gradh, grad f) ∆(fh) = f∆h+ 2g(gradh, grad f) + h∆f. The divergence theorem now gives us Green’s Formulae: Theorem. For f, h ∈ C∞(M) with at least one of f and h compactly sup- ported: ∫ M h∆f dV = − ∫ M g(grad f, gradh) dV∫ M h∆f dV = ∫ M f∆h dV. 2.3.3 Boundary terms Supposed that M is oriented and that Ω ⊂M is an open subset with smooth boundary ∂Ω. Thus ∂Ω is a smooth manifold with 1. a Riemannian metric inherited via (∂Ω)m ⊂Mm; 12 2. a Riemannian measure dA; 3. a unique outward-pointing normal unit vector field ν. With these ingredients, one has: Divergence Theorem II. Any compactly supported X on M has∫ Ω divX dV = ∫ ∂Ω g(X, ν) dA and so Green’s Formulae: Theorem. For f, h ∈ C∞(M) with at least one of f and h compactly sup- ported:∫ Ω h∆f + 〈grad f, gradh〉 dV = ∫ ∂Ω h〈ν, grad f〉 dA∫ Ω h∆f − ∫ Ω f∆h dV = ∫ ∂Ω h〈ν, grad f〉 dA− ∫ ∂Ω f〈ν, gradh〉 dA where we have written 〈 , 〉 for g( , ). In particular ∫ Ω ∆f dV = ∫ ∂Ω νf dV. 3 Geodesics and curvature In the classical geometry of Euclid, a starring role is played by the straight lines. Viewed as paths of shortest length between two points, these may be generalised to give a distinguished family of paths, the geodesics, on any Riemannian manifold. Geodesics provide a powerful tool to probe the geometry of Riemannian manifolds. Notation. Let (M, g) be a Riemannian manifold. For ξ, η ∈Mm, write g(ξ, η) = 〈ξ, η〉, √ g(ξ, ξ) = |ξ|. 3.1 (M, g) is a metric space A piece-wise C1 path γ : [a, b]→M has length L(γ): L(γ) = ∫ b a |γ′(t)|dt. 13 Exercise. The length of a path is invariant under reparametrisation. Recall that M is connected and so3 path-connected. For p, q ∈M , set d(p, q) = inf{L(γ) : γ : [a, b]→M is a path with γ(a) = p, γ(b) = q}. One can prove: • (M,d) is a metric space. • The metric space topology coincides with the original topology on M . The key points here are the definiteness of d and the assertion about the topologies. For this, it is enough to work in a precompact open subset of a chart U where one can prove the existence of K1,K2 ∈ R such that K1 ∑ 1≤i≤n ξ2i ≤ ∑ i,j gijξiξj ≤ K2 ∑ 1≤i≤n ξ2i . From this, one readily sees that, on such a subset, d is equivalent to the Euclidean metric on U . 3.2 Parallel vector fields and geodesics Let c : I → M be a path. Recall the pull-back connection c−1∇ on the space Γ(c−1TM) of vector fields along c. This connection gives rise to a differential operator ∇t : Γ(c−1TM)→ Γ(c−1TM) by ∇tY = (c−1∇)∂1Y where ∂1 is the coordinate vector field on I. Note that since ∇ is metric, we have 〈X,Y 〉′ = 〈∇tX,Y 〉+ 〈X,∇tY 〉, for X,Y ∈ Γ(c−1TM). Definition. X ∈ Γ(c−1TM) is parallel if ∇tX = 0. The existence and uniqueness results for linear ODE give: 3Manifolds are locally path-connected! 14 Proposition. For c : [a, b] → M and U0 ∈ Mc(a), there is unique parallel vector field U along c with U(a) = U0. If Y1, Y2 are parallel vector fields along c, then all 〈Yi, Yj〉 and, in particular, |Yi| are constant. Definition. γ : I →M is a geodesic if γ′ is parallel: ∇tγ′ = 0. It is easy to prove that, for a geodesic γ: • |γ′| is constant. • If γ is a geodesic, so is t 7→ γ(st) for s ∈ R. The existence and uniqueness results for ODE give: 1. For ξ ∈ Mm, there is a maximal open interval Iξ ⊂ R on which there is a unique geodesic γξ : Iξ →M such that γξ(0) = m γ′ξ(0) = ξ. 2. (t, ξ) 7→ γξ(t) is a smooth map Iξ ×Mm →M . 3. γsξ(t) = γξ(st). Let us collect some examples: 1. M = Rn with its canonical metric. The geodesic equation reduces to: d2γ dt2 = 0 and we conclude that geodesics are straight lines. 2. M = Sn and ξ is a unit vector in Mm = m⊥. Contemplate reflection in the 2-plane spanned by m and ξ: this induces a map Φ : Sn → Sn which preserves the metric and so ∇ also while it fixes m and ξ. Thus, if γ is a geodesic so is Φ ◦ γ and the uniqueness part of the ODE yoga forces Φ ◦ γξ = γξ. Otherwise said, γξ lies in the plane spanned by m and ξ and so lies on a great circle. 15 To get further, recall that |γ′ξ| = |ξ| = 1 which implies: γξ(t) = (cos t)m+ (sin t)ξ. A similar argument shows that the unique parallel vector field U along γξ with U(0) = η ⊥ ξ is given by U ≡ η. 3. M = Dn with the hyperbolic metric and ξ is a unit vector in M0 ∼= Rn. Again, symmetry considerations force γξ to lie on the straight line through 0 in the direction of ξ and then |γ′ξ| = 1 gives: γξ(t) = (2 tanh t/2)ξ. Similarly, the parallel vector field along γξ with U(0) = η ⊥ ξ is given by U(t) = 1 cosh2 t/2 η. 3.3 The exponential map 3.3.1