Chapter 6 Riemannian Manifolds and Connections 6.1 Riemannian Metrics Fortunately, the rich theory of vector spaces endowed with aEuclideaninnerproductcan,toagreatextent,belifted to various bundles associated with a manifold. The notion of local (and global) frame plays an important technical role. Definition 6.1.1 Let M be an n-dimensional smooth manifold. For any open subset, U M,ann-tuple of ⊆ vector fields, (X1,...,Xn), over U is called a frame over U iff(X1(p),...,Xn(p)) is a basis of the tangent space, T M,foreveryp U.IfU = M,thentheX are global p ∈ i sections and (X1,...,Xn)iscalledaframe (of M). 455 456 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS The notion of a frame is due to Elie´ Cartan who (after Darboux) made extensive use of them under the name of moving frame (and the moving frame method). Cartan’s terminology is intuitively clear: As a point, p, moves in U,theframe,(X1(p),...,Xn(p)), moves from fibre to fibre. Physicists refer to a frame as a choice of local gauge. If dim(M)=n,thenforeverychart,(U,ϕ), since 1 n dϕ− : R T M is a bijection for every p U,the ϕ(p) → p ∈ n-tuple of vector fields, (X1,...,Xn), with 1 Xi(p)=dϕϕ−(p)(ei), is a frame of TM over U,where n (e1,...,en)isthecanonicalbasisofR . The following proposition tells us when the tangent bun- dle is trivial (that is, isomorphic to the product, M Rn): × 6.1. RIEMANNIAN METRICS 457 Proposition 6.1.2 The tangent bundle, TM,ofa smooth n-dimensional manifold, M,istrivialiffit possesses a frame of global sections (vector fields de- fined on M). As an illustration of Proposition 6.1.2 we can prove that the tangent bundle, TS1,ofthecircle,istrivial. Indeed, we can find a section that is everywhere nonzero, i.e. anon-vanishingvectorfield,namely X(cos θ,sin θ)=( sin θ, cos θ). − The reader should try proving that TS3 is also trivial (use the quaternions). However, TS2 is nontrivial, although this not so easy to prove. More generally, it can be shown that TSn is nontrivial for all even n 2. It can even be shown that S1, S3 and S7 are the only≥ spheres whose tangent bundle is trivial. This is a rather deep theorem and its proof is hard. 458 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS Remark: Amanifold,M,suchthatitstangentbundle, TM,istrivialiscalledparallelizable. We now define Riemannian metrics and Riemannian man- ifolds. Definition 6.1.3 Given a smooth n-dimensional man- ifold, M,aRiemannian metric on M (or TM) is a family, ( , p)p M ,ofinnerproductsoneachtangent '− −( ∈ space, TpM,suchthat , p depends smoothly on p, which means that for every'− chart,−( ϕ : U Rn,forevery α α → frame, (X1,...,Xn), on Uα,themaps p X (p),X (p) ,pU , 1 i, j n )→ ' i j (p ∈ α ≤ ≤ are smooth. A smooth manifold, M,withaRiemannian metric is called a Riemannian manifold. 6.1. RIEMANNIAN METRICS 459 If dim(M)=n,thenforeverychart,(U,ϕ), we have the 1 frame, (X1,...,Xn), over U,withXi(p)=dϕϕ−(p)(ei), n where (e1,...,en)isthecanonicalbasisofR .Sinceev- n ery vector field over U is a linear combination, i=1 fiXi, for some smooth functions, fi: U R,theconditionof Definition 6.1.3 is equivalent to the→ fact that the! maps, 1 1 p dϕ− (e ),dϕ− (e ) ,pU, 1 i, j n, )→ ' ϕ(p) i ϕ(p) j (p ∈ ≤ ≤ are smooth. If we let x = ϕ(p), the above condition says that the maps, 1 1 x dϕ− (ei),dϕ− (ej) 1 ,x ϕ(U), 1 i, j n, )→ ' x x (ϕ− (x) ∈ ≤ ≤ are smooth. If M is a Riemannian manifold, the metric on TM is often denoted g =(gp)p M .Inachart,usinglocalcoordinates, we often use the notation∈ g = g dx dx or simply ij ij i ⊗ j g = gijdxidxj,where ij ! ! ∂ ∂ gij(p)= , . ∂xi ∂xj "# $p # $p%p 460 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS For every p U,thematrix,(gij(p)), is symmetric, pos- itive definite.∈ The standard Euclidean metric on Rn,namely, g = dx2 + + dx2, 1 ··· n makes Rn into a Riemannian manifold. Then, every submanifold, M,ofRn inherits a metric by restricting the Euclidean metric to M. n 1 For example, the sphere, S − ,inheritsametricthat n 1 makes S − into a Riemannian manifold. It is a good exercise to find the local expression of this metric for S2 in polar coordinates. AnontrivialexampleofaRiemannianmanifoldisthe Poincar´eupper half-space,namely,theset H = (x, y) R2 y>0 equipped with the metric { ∈ | } dx2 + dy2 g = . y2 6.1. RIEMANNIAN METRICS 461 Awaytoobtainametriconamanifold,N,istopull- back the metric, g,onanothermanifold,M,alongalocal diffeomorphism, ϕ: N M. → Recall that ϕ is a local diffeomorphism iff dϕ : T N T M p p → ϕ(p) is a bijective linear map for every p N. ∈ Given any metric g on M,ifϕ is a local diffeomorphism, we define the pull-back metric, ϕ∗g,onN induced by g as follows: For all p N,forallu, v T N, ∈ ∈ p (ϕ∗g)p(u, v)=gϕ(p)(dϕp(u),dϕp(v)). We need to check that (ϕ∗g)p is an inner product, which is very easy since dϕp is a linear isomorphism. Our map, ϕ,betweenthetwoRiemannianmanifolds (N,ϕ∗g)and(M,g)isalocalisometry,asdefinedbe- low. 462 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS Definition 6.1.4 Given two Riemannian manifolds, (M1,g1)and(M2,g2), a local isometry is a smooth map, ϕ: M M ,suchthatdϕ : T M T M is an 1 → 2 p p 1 → ϕ(p) 2 isometry between the Euclidean spaces (TpM1, (g1)p)and (T M , (g ) ), for every p M ,thatis, ϕ(p) 2 2 ϕ(p) ∈ 1 (g1)p(u, v)=(g2)ϕ(p)(dϕp(u),dϕp(v)), for all u, v TpM1 or, equivalently, ϕ∗g2 = g1.More- over, ϕ is an∈ isometry iffit is a local isometry and a diffeomorphism. The isometries of a Riemannian manifold, (M,g), form a group, Isom(M,g), called the isometry group of (M,g). An important theorem of Myers and Steenrod asserts that the isometry group, Isom(M,g), is a Lie group. 6.1. RIEMANNIAN METRICS 463 Given a map, ϕ: M M ,andametricg on M ,in 1 → 2 1 1 general, ϕ does not induce any metric on M2. However, if ϕ has some extra properties, it does induce ametriconM2.ThisisthecasewhenM2 arises from M1 as a quotient induced by some group of isometries of M1. For more on this, see Gallot, Hulin and Lafontaine [?], Chapter 2, Section 2.A. Now, because a manifold is paracompact (see Section 4.6), a Riemannian metric always exists on M.Thisis aconsequenceoftheexistenceofpartitionsofunity(see Theorem 4.6.5). Theorem 6.1.5 Every smooth manifold admits a Rie- mannian metric. 464 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS 6.2 Connections on Manifolds Given a manifold, M,ingeneral,foranytwopoints, p, q M,thereisno“natural”isomorphismbetween ∈ the tangent spaces TpM and TqM. Given a curve, c:[0, 1] M,onM as c(t)moveson → M,howdoesthetangentspace,Tc(t)M change as c(t) moves? n n If M = R ,thenthespaces,Tc(t)R ,arecanonically n n n isomorphic to R and any vector, v Tc(0)R ∼= R ,is simply moved along c by parallel transport∈ ,thatis,at n c(t), the tangent vector, v,alsobelongstoTc(t)R . However, if M is curved, for example, a sphere, then it is not obvious how to “parallel transport” a tangent vector at c(0) along a curve c. 6.2. CONNECTIONS ON MANIFOLDS 465 Awaytoachievethisistodefinethenotionofparallel vector field along a curve and this, in turn, can be defined in terms of the notion of covariant derivative of a vector field. Assume for simplicity that M is a surface in R3. Given any two vector fields, X and Y defined on some open sub- set, U R3,foreveryp U,thedirectional derivative, ⊆ ∈ DXY (p),ofY with respect to X is defined by Y (p + tX(p)) Y (p) DXY (p)=lim − . t 0 → t If f: U R is a differentiable function on U,forevery p U,the→ directional derivative, X[f](p) (or X(f)(p)), of∈f with respect to X is defined by f(p + tX(p)) f(p) X[f](p)=lim − . t 0 → t We know that X[f](p)=df p(X(p)). 466 CHAPTER 6. RIEMANNIAN MANIFOLDS AND CONNECTIONS It is easily shown that DXY (p)isR-bilinear in X and Y , is C∞(U)-linear in X and satisfies the Leibnitz derivation rule with respect to Y ,thatis: Proposition 6.2.1 The directional derivative of vec- tor fields satisfies the following properties: DX1+X2Y (p)=DX1Y (p)+DX2Y (p) DfXY (p)=fDXY (p) DX(Y1 + Y2)(p)=DXY1(p)+DXY2(p) DX(fY)(p)=X[f](p)Y (p)+f(p)DXY (p), for all X, X ,X ,Y,Y ,Y X(U) and all f C∞(U). 1 2 1 2 ∈ ∈ Now, if p U where U M is an open subset of M,for ∈ ⊆ any vector field, Y ,definedonU (Y (p) TpM,forall p U), for every X T M,thedirectionalderivative,∈ ∈ ∈ p DXY (p), makes sense and it has an orthogonal decom- position, D Y (p)= Y (p)+(D ) Y (p), X ∇X n X where its horizontal (or tangential) component is Y (p) T M and its normal component is (D ) Y (p). ∇X ∈ p n X 6.2. CONNECTIONS ON MANIFOLDS 467 The component, Y (p), is the covariant derivative of ∇X Y with respect to X TpM and it allows us to define the covariant derivative∈ of a vector field, Y X(U), with respect to a vector field, X X(M), on M∈.