LECTURE4:THERIEMANNIANMEASURE1.THERIEMANNIANMEASUREITheRiemannian volumeintangent space.Not only a Riemannian metric g (as an infinitesimal distance, i.e. a distancedefined in each tangent space) on M gives rise to a canonical metric structure onM,but also it defines a canonical measure structure (or to be more precise, avolume density) on M through an “infinitesimal volume"(i.e.volume defined ineach tangent space). The idea is standard: as in multi-variable calculus, to definethe volume or integrate a function over M, one simply start with a coordinate chart,using which one can divide M into small coordinate pieces, and then approximateeach small piece [(rl,...,rm) [a' ≤ r≤ a' + hi by the parallelepiped in T,M(wherep= (rl,...,rm))generated by (h'i,...,hmom).Now the problem is reduces to: how do we define a volume of a parallelepipedin a finite dimensional inner product space? Well, one can always define the vol-ume of a unit cube to be 1 (here we use not only the lengths of vectors, but also the anglesbetween vectors), and then use multi-linearity to extend the definition to more gen-eral parallelepipeds.Soto computethevolumeof theparallelepipedgeneratedbyOi,O2,...,Om,we start with any an orthonormal basis ei,...,em of (T,M,gp), anddefine the volume of theparallelotopegenerated by ei,...,em to beV,(ei,e2,...,em)=1.Thenwewrite,=d,ej,which impliesVp(α1,02,... , Om) = [det(α)l.For simplicitywedenoteA=(a).Fromtheobservationgj = g(O,0) = g(dfex,djel) =atd, = (AAT)gwe conclude (gis) = AAT, and thus the “infinitesimal volume" we are calculating isVp(α1, 02,..., om) = [det(α)/ = VG,where G=det(gi).Remark.Alternatively,one can define V,(Oi,2,...,Om)as"thelength of thevector,A2A...Am in the spacemT,M"(withrespect to the induced metric on tensorsthat we introduced in Lecture 2), and similar computation yields the same result
LECTURE 4: THE RIEMANNIAN MEASURE 1. The Riemannian measure ¶ The Riemannian volume in tangent space. Not only a Riemannian metric g (as an infinitesimal distance, i.e. a distance defined in each tangent space) on M gives rise to a canonical metric structure on M, but also it defines a canonical measure structure (or to be more precise, a volume density) on M through an “infinitesimal volume” (i.e. volume defined in each tangent space). The idea is standard: as in multi-variable calculus, to define the volume or integrate a function over M, one simply start with a coordinate chart, using which one can divide M into small coordinate pieces, and then approximate each small piece {(x 1 , · · · , xm) |a i ≤ x i ≤ a i + h i} by the parallelepiped in TpM (where p = (x 1 , · · · , xm)) generated by (h 1∂1, · · · , hm∂m). Now the problem is reduces to: how do we define a volume of a parallelepiped in a finite dimensional inner product space? Well, one can always define the volume of a unit cube to be 1 (here we use not only the lengths of vectors, but also the angles between vectors), and then use multi-linearity to extend the definition to more general parallelepipeds. So to compute the volume of the parallelepiped generated by ∂1, ∂2, · · · , ∂m, we start with any an orthonormal basis e1, · · · , em of (TpM, gp), and define the volume of the parallelotope generated by e1, · · · , em to be Vp(e1, e2, · · · , em) = 1. Then we write ∂i = a j i ej , which implies Vp(∂1, ∂2, · · · , ∂m) = | det(a j i )|. For simplicity we denote A = (a j i ). From the observation gij = g(∂i , ∂j ) = g(a k i ek, al j el) = X k a k i a k j = (AAT )ij , we conclude (gij ) = AAT , and thus the “infinitesimal volume” we are calculating is Vp(∂1, ∂2, · · · , ∂m) = | det(a j i )| = √ G, where G = det(gij ). Remark. Alternatively, one can define Vp(∂1, ∂2, · · · , ∂m) as “the length of the vector ∂1∧∂2∧· · ·∧∂m in the space ⊗mTpM” (with respect to the induced metric on tensors that we introduced in Lecture 2), and similar computation yields the same result. 1
2LECTURE4:THERIEMANNIANMEASUREI Integrals of compactly supported continuous functions.Now let (M,g)be a Riemannian manifold.We start with a continuous functionf with compact support, so that supp(f) is contained in one chart (o, U,V). Asmotivated by the previous computation, we may definefdVg :(fvG)op-1 dr...-drmwhere drl.. -drm the Lebesgue measure on Rm.Lemma 1.l. The definition above is independent of the choices of coordinate chartscontaining supp(f)Proof. Let (@,U, V) be another coordinate chart containing supp(f), on which thecoordinates are denoted by y', .. , ym. Then as we have seen in Lecture 2(gig) = JT(gu)J,where J = (%) is the Jacobian of the map o-1. As a consequence, we getVG(p) = VG(p) I det(J((p)Ifor p = p-1(r) = -1(y), and thus by change of variables in Rm,VGog-1dyl..-dym = VGogp-1(βo p-1) det(J)ldrl...drm = VGop-idrl..-drm.口The conclusion follows.Of course in general, evenif f is compactly supported, one cannot assumethatsupp(f) is contained in one single chart. However, one can extend the above defi-nition to general f e Ce(M) easily by using partition of unity:Let [(βa,Ua,V))be a system of locally finite coordinate charts that cover M, with local coordinates[ca, ..., cm] on each Ua, and let (pa] be a partition of unity subordinate to theopen coveringUa].ThenwedefinefdVg:=/(fpaVGa)(a)-idr...dam,a(U.Note that by locally finiteness of Uaand compactness of supp(f), the sum is in facta finite sum. Moreover, if ((p,Up, Vo)) is another atlas, the by Lemma 1.1,(fpappVGa)o(pa)-1de..-drm =(fpapVGB)o(pp)-1dcg..dcmPa(UanUg)JpB(UanUg)since both sides equal to J PapefdVg, which implies(fpaVGa)o(pa)-'dr..dam -/(fpgVGP)o(pp)-1dcg...dacm>B(UB)In other words, JM fdVg is well-defined for any f e Ce(M)
2 LECTURE 4: THE RIEMANNIAN MEASURE ¶ Integrals of compactly supported continuous functions. Now let (M, g) be a Riemannian manifold. We start with a continuous function f with compact support, so that supp(f) is contained in one chart (φ, U, V ). As motivated by the previous computation, we may define Z M f dVg := Z V (f √ G) ◦ φ −1 dx1 · · · dxm, where dx1 · · ·dxm the Lebesgue measure on R m. Lemma 1.1. The definition above is independent of the choices of coordinate charts containing supp(f). Proof. Let ( ˜φ, U, e Ve) be another coordinate chart containing supp(f), on which the coordinates are denoted by y 1 , · · · , ym. Then as we have seen in Lecture 2, (gij ) = J T (˜gkl)J, where J = ( ∂yi ∂xj ) is the Jacobian of the map φe ◦ φ −1 . As a consequence, we get p G(p) = q Ge(p) | det(J(φ(p)))| for p = φ −1 (x) = ˜φ −1 (y), and thus by change of variables in R m, q Ge◦φ˜ −1dy1 · · ·dym = q Ge◦φ˜ −1 ( ˜φ ◦ φ−1 ) |det(J)|dx1 · · ·dxm = p G◦φ−1dx1 · · ·dxm. The conclusion follows. □ Of course in general, even if f is compactly supported, one cannot assume that supp(f) is contained in one single chart. However, one can extend the above definition to general f ∈ Cc(M) easily by using partition of unity: Let {(φα, Uα, Vα)} be a system of locally finite coordinate charts that cover M, with local coordinates {x 1 α , · · · , xm α } on each Uα, and let {ρα} be a partition of unity subordinate to the open covering {Uα}. Then we define Z M f dVg := X α Z φα(Uα) (f ρα √ Gα) ◦ (φα) −1 dx1 α · · · dxm α , Note that by locally finiteness of Uα and compactness of supp(f), the sum is in fact a finite sum. Moreover, if {( ˜φβ, Ueβ, Veβ)} is another atlas, the by Lemma 1.1, Z φα(Uα∩Ueβ) (f ραρ˜β √ Gα)◦(φα) −1 dx1 α · · ·dxm α = Z φ˜β(Uα∩Ueβ) (f ραρ˜β √ Gβ)◦(φβ) −1 dx1 β · · ·dxm β since both sides equal to R M ραρ˜βf dVg, which implies X α Z φα(Uα) (f ρα √ Gα)◦(φα) −1 dx1 α · · · dxm α = X β Z φβ(Uβ) (f ρβ √ Gβ)◦(φβ) −1 dx1 β · · · dxm β . In other words, R M f dVg is well-defined for any f ∈ Cc(M)
3LECTURE4:THERIEMANNIANMEASURET The Riemannian measure.Since manifolds are always locally compact and Hausdorff, and since the linearfunctionalμ:Ce(M)→R, f →μ(f)= / fdVgis positive (i.e. f ≥0 implies μ(f)≥0), by Riesz representation theorem,μ gives riseto a unique Radon measure on M. Now one can further extend the integral to moregeneral functions using the standard machinery developed in real analysis:. first define the (upper) integral of a lower semi-continuous positive functionf to be the supremum of integrals of compactly-supported functions that areno more than f,.then define the (upper) integral of a positive function f as the infimum ofthe (upper) integral of all lower semi-continuous positive function f that aregreater than f,.a function f is said to be integrable if there exists a sequence gn in Ce(M) so that the (upper) integrals of the sequence [gn - fl converge to 0.As usual we denotethe space of integrablefunctions as L'(M,g),whichby definitionis the completion of Ce(M) with respect to suitable norm.As usual, for any 1 ≤ p < oo one can define the Lp norm on Coo via/(.ifpdv,)Ilfl p :=and define LP(M,g) to be the completion of Co under the Lp norm. Similarlyone can define L(M,g). It is not hard to extend the theory to complex-valuedfunctions.In the special case p =2, one can define an inner product structure onL?(M,g) by(fi, f2) r2 :=fifadvgwhich make L?(M,g) into a Hilbert space.One can also talk about the volume of any Borel set (or more generally,mea-surable subsets) A in M, which is defined to beVol(A) =XAdVgRemark. In the above definition, we don't assume M to be oriented or compact.Whatwereally get isa volume density,which, on a local chart, can be written asdVg=VGop-1del...damWe will call dVol the Riemannian volume element (or volume density) on (M,g)
LECTURE 4: THE RIEMANNIAN MEASURE 3 ¶ The Riemannian measure. Since manifolds are always locally compact and Hausdorff, and since the linear functional µ : Cc(M) → R, f 7→ µ(f) = Z M f dVg is positive (i.e. f ≥ 0 implies µ(f) ≥ 0), by Riesz representation theorem, µ gives rise to a unique Radon measure on M. Now one can further extend the integral to more general functions using the standard machinery developed in real analysis: • first define the (upper) integral of a lower semi-continuous positive function f to be the supremum of integrals of compactly-supported functions that are no more than f, • then define the (upper) integral of a positive function f as the infimum of the (upper) integral of all lower semi-continuous positive function f that are greater than f, • a function f is said to be integrable if there exists a sequence gn in Cc(M) so that the (upper) integrals of the sequence |gn − f| converge to 0. As usual we denote the space of integrable functions as L 1 (M, g), which by definition is the completion of Cc(M) with respect to suitable norm. As usual, for any 1 ≤ p < ∞ one can define the L p norm on C ∞ c via ∥f∥Lp := Z M |f| p dVg 1/p , and define L p (M, g) to be the completion of C ∞ c under the L p norm. Similarly one can define L ∞(M, g). It is not hard to extend the theory to complex-valued functions. In the special case p = 2, one can define an inner product structure on L 2 (M, g) by ⟨f1, f2⟩L2 := Z M f1 ¯f2dVg which make L 2 (M, g) into a Hilbert space. One can also talk about the volume of any Borel set (or more generally, measurable subsets) A in M, which is defined to be Vol(A) = Z M χAdVg Remark. In the above definition, we don’t assume M to be oriented or compact. What we really get is a volume density, which, on a local chart, can be written as dVg = √ G ◦ φ −1 dx1 · · · dxm . We will call dVol the Riemannian volume element (or volume density) on (M, g)
4LECTURE4:THERIEMANNIANMEASURERemark. In the special case where M is oriented, then we may choose an orientation-compatiblecoordinatepatchnear eachpoint,anddefine(locallyon eachchart)wg=VGdalA...Λdarm.One can check that wis a well-defined global volumeform on M,which is calledthe Riemannian volume form for the oriented Riemannian manifold (M,g).Remark. Suppose (M,g) is an m-dimensional Riemannian manifold, and S an r-dimensional submanifold of M,wherer <m.Then theRiemannian submanifoldmetric gs := t*g on M gives a natural measure (an r-dimensional volume density)on S. Here are two special cases:. If : I → M is a simple smooth curve, then with respect to the coordinatest (from the parametrization), we have g= g(ot, at)dt dt =[(t)2dt dt,and thus the induced 1-dimensional volume density (i.e. length density) onis simply ildt,which is exactly what we used to calculate thelength of. If M is a smooth manifold with boundary, in which case the boundary oM isa smooth submanifold of dimension m-l, then onegets a natural Riemann-ian submanifold metric and thus a volume density on oM. In this case thevolume density on oM is usually called a surface density (or hypersurfacedensity)and will be denoted by dSgI The change of variable formula.By using the standard change of variable formula for the Lebesgue measure inRm, together with a partition of unity argument, one can easily prove the followingProposition 1.2 (Change of variables in Riemannian setting). Let :M → N bea diffeomorphism, and h a Riemannian metric on N. ThenfopdVoh=f dVh, Vf L'(N,h)In particular, we see isometries preserve the Riemannian volume densities.Asanotherconsequence,supposedimM≤dimN,:M→Nisan embedding,and t : p(M) -→ N is the inclusion map. Let g be a Riemannian metric on M andh be a Riemannian metric on N, thendVghdVgf dVeh,Vf eL'(N,h),fopdVge is the Radon-Nikodyn derivative of the two corresponding Riemannianwheremeasures on M (which are by definition o-finite measures). In particular, one mayfind the area of g(M) (or integrals over (M)) in the target space by doing computationsin the source space M. It is a very special case of the so-called area formula inINote that it makes no sense to write an expression like Jy f oJdetdol dVg even if is adiffeomorphism, since dp is a linear map between different vector spaces
4 LECTURE 4: THE RIEMANNIAN MEASURE Remark. In the special case where M is oriented, then we may choose an orientationcompatible coordinate patch near each point, and define (locally on each chart) ωg = √ Gdx1 ∧ · · · ∧ dxm. One can check that ωg is a well-defined global volume form on M, which is called the Riemannian volume form for the oriented Riemannian manifold (M, g). Remark. Suppose (M, g) is an m-dimensional Riemannian manifold, and S an rdimensional submanifold of M, where r < m. Then the Riemannian submanifold metric gS := ι ∗ g on M gives a natural measure (an r-dimensional volume density) on S. Here are two special cases: • If γ : I → M is a simple smooth curve, then with respect to the coordinates t (from the parametrization), we have gγ = g(∂t , ∂t)dt ⊗ dt = |γ˙(t)| 2dt ⊗ dt, and thus the induced 1-dimensional volume density (i.e. length density) on γ is simply |γ˙ |dt, which is exactly what we used to calculate the length of γ. • If M is a smooth manifold with boundary, in which case the boundary ∂M is a smooth submanifold of dimension m−1, then one gets a natural Riemannian submanifold metric and thus a volume density on ∂M. In this case the volume density on ∂M is usually called a surface density (or hypersurface density) and will be denoted by dSg. ¶ The change of variable formula. By using the standard change of variable formula for the Lebesgue measure in R m, together with a partition of unity argument, one can easily prove the following Proposition 1.2 (Change of variables in Riemannian setting). Let φ : M → N be a diffeomorphism, and h a Riemannian metric on N. Then Z M f ◦ φ dVφ∗h = Z N f dVh, ∀f ∈ L 1 (N, h). In particular, we see isometries preserve the Riemannian volume densities. As another consequence, suppose dim M ≤ dim N, φ : M → N is an embedding, and ι : φ(M) → N is the inclusion map. Let g be a Riemannian metric on M and h be a Riemannian metric on N, then 1 Z M f ◦ φ dVφ∗h dVg dVg = Z φ(M) f dVι ∗h, ∀f ∈ L 1 (N, h), where dVφ∗h dVg is the Radon-Nikodyn derivative of the two corresponding Riemannian measures on M (which are by definition σ-finite measures). In particular, one may find the area of φ(M) (or integrals over φ(M)) in the target space by doing computations in the source space M. It is a very special case of the so-called area formula in 1Note that it makes no sense to write an expression like R M f ◦ φ |det dφ| dVg even if φ is a diffeomorphism, since dφ is a linear map between different vector spaces
5LECTURE4:THERIEMANNIANMEASUREgeometric measure theory where is only supposed to be Lipschitz and need notbeinjective,and themeasuresencountered arereplaced bythe Hausdorff measureThere is a “dual" version of the area formula above, known as the co-area for-mula, in which, with the help of a map : M → N with dim M ≥ dim N, one coulduse integrals over level sets p-1(g) in target space to compute integrals over thesource space M. In the very general version of co-area formula in geometric mea-suretheory,is onlysupposed tobea Lipschitzmap,and peopleusetheHausdorfmeasures. In what follows we will prove a simplest version of co-area formula (forN=R)that is already very useful in Riemannian geometry.To state the theorem,we need the concept of gradient vector fields associated to a function.I The gradient.Let (M,g)be a Riemannian manifold. For any smooth function f on M, thedifferential df is a smooth 1-form on M.By using the musical isomorphism:T*M→TM,wewill getasmoothvectorfield onM:Definition 1.3. The gradient vector field of f is f =#(df),It is not hard to find out vf in local charts:By definition, Vf is the vectorfield so that for any vector field X = Xai,g(Vf, X) = df(X) = Xf = Xio,f.It follows that locallyvf=giafa,In particular, for g = go in Rm, we get the ordinary gradient of f.As in multivariable calculus, the gradient vector field of a function is alwaysperpendicular to its regular level sets:Lemma 1.4. Suppose f is a smooth function on M and c is a regular value of fThen the gradient vector field Vf is perpendicular to the level set f-1(t)Proof.Since c is a regular value, by the regular level set theorem, f-1(c) is a smoothsubmanifold of M. Let X be a vector field tangent to f-1(c). Then we learned frommanifold theory that X f = 0 on f-1(c). It followsg(Vf,X) =Xf = 0口on f-1(c). So Vf is perpendicular to f-1(c).I The Coarea formula: a simple version.Fix a smooth function u ECo(M) and let2t := u-l((-00,t),It := u-l(t)For any regular value t of u, It is a smooth submanifold of dimension m -1 in M.BySard's theorem,critical valuesof uform ameasurezero set inR (and thus canbe ignored in the integration below). Now we can prove
LECTURE 4: THE RIEMANNIAN MEASURE 5 geometric measure theory where φ is only supposed to be Lipschitz and need not be injective, and the measures encountered are replaced by the Hausdorff measure. There is a “dual” version of the area formula above, known as the co-area formula, in which, with the help of a map φ : M → N with dim M ≥ dim N, one could use integrals over level sets φ −1 (q) in target space to compute integrals over the source space M. In the very general version of co-area formula in geometric measure theory, φ is only supposed to be a Lipschitz map, and people use the Hausdorff measures. In what follows we will prove a simplest version of co-area formula (for N = R) that is already very useful in Riemannian geometry. To state the theorem, we need the concept of gradient vector fields associated to a function. ¶ The gradient. Let (M, g) be a Riemannian manifold. For any smooth function f on M, the differential df is a smooth 1-form on M. By using the musical isomorphism ♯ : T ∗M → TM, we will get a smooth vector field on M: Definition 1.3. The gradient vector field of f is ∇f = ♯(df). It is not hard to find out ∇f in local charts: By definition, ∇f is the vector field so that for any vector field X = Xi∂i , g(∇f, X) = df(X) = Xf = X i ∂if. It follows that locally ∇f = g ij∂if ∂j . In particular, for g = g0 in R m, we get the ordinary gradient of f. As in multivariable calculus, the gradient vector field of a function is always perpendicular to its regular level sets: Lemma 1.4. Suppose f is a smooth function on M and c is a regular value of f. Then the gradient vector field ∇f is perpendicular to the level set f −1 (t). Proof. Since c is a regular value, by the regular level set theorem, f −1 (c) is a smooth submanifold of M. Let X be a vector field tangent to f −1 (c). Then we learned from manifold theory that Xf = 0 on f −1 (c). It follows g(∇f, X) = Xf = 0 on f −1 (c). So ∇f is perpendicular to f −1 (c). □ ¶ The Coarea formula: a simple version. Fix a smooth function u ∈ C ∞(M) and let Ωt := u −1 ((−∞, t)), Γt := u −1 (t). For any regular value t of u, Γt is a smooth submanifold of dimension m − 1 in M. By Sard’s theorem, critical values of u form a measure zero set in R (and thus can be ignored in the integration R R below). Now we can prove
6LECTURE4:THERIEMANNIANMEASURETheorem 1.5 (The co-area formula, a simple version).Let (M,g) be a Riemannianmanifold. For any regular value t of u, let gt be the induced Riemannian metric onIt.and denote the corresponding Riemannian volume density on It by dSt.Thenfor any integrable function f on M, one has(flvuldvg=/(/ fds.)dt1Proof.First note that if we let C be the set of critical points of u, then C is closed.It follows that MC is an open submanifold in M, and obviouslyfiVuldV,=fivuldvgSo we may replace M by M C without changing both sides. In other words, wemayassumeuadmitsnocriticalpointonMNow consider the vector fieldVuX/Vul2on M.By Lemma 1.4, X is perpendicular toT,Fat any q EF.for any c. Let ptbe the (local) flow generated by X. Then by definition,du(p(a) = da(X(g(0) = (Vu, X)e() = 1. It follows that if q E Fe, then Pt(q) e Ie+t for t small enough. Now we choose aneighborhood A of q in Fe so thatb : (-e,e) ×A-→M, (y,t) -pt(y)is a diffeomorphism onto an open subset U = ((-e,e) x A) in M. By shrinking Aifnecessary,we may supposeAis a coordinatepatch onI。and let y',...,ym-1 becorresponding coordinate functions.Then [t,y',...,ym-1) form a set of coordinatefunctions on U. With respect to these coordinates, and in view of the facts t = Xand X I Oy for all i, the Riemannian metric g has the formg = (X, X)dt @ dt + hijdy' dy,where hj = g(y, y), Since (X,X)=p, the volume density/det (hij)dtdy* ...dym-1dVgdtdSt[Vul(VulSo we conclude that for any pE Ce(U),pf Vdet Gtdtdy'...dym-1 pflVuldVg=pfdStX口Now the conclusion follows from a standard partition of unity argument.As a corollary, we get
6 LECTURE 4: THE RIEMANNIAN MEASURE Theorem 1.5 (The co-area formula, a simple version). Let (M, g) be a Riemannian manifold. For any regular value t of u, let gt be the induced Riemannian metric on Γt. and denote the corresponding Riemannian volume density on Γt by dSt. Then for any integrable function f on M, one has Z M f|∇u|dVg = Z R Z Γt f dSt dt. Proof. First note that if we let C be the set of critical points of u, then C is closed. It follows that M \ C is an open submanifold in M, and obviously Z M f|∇u|dVg = Z M\C f|∇u|dVg. So we may replace M by M \ C without changing both sides. In other words, we may assume u admits no critical point on M. Now consider the vector field X = ∇u |∇u| 2 on M. By Lemma 1.4, X is perpendicular to TqΓc at any q ∈ Γc for any c. Let φt be the (local) flow generated by X. Then by definition, d dtu(φt(q)) = du(X(φt(q))) = ⟨∇u, X⟩φt(q) = 1. It follows that if q ∈ Γc, then φt(q) ∈ Γc+t for t small enough. Now we choose a neighborhood A of q in Γc so that ψ : (−ε, ε) × A → M, (y, t) 7→ φt(y) is a diffeomorphism onto an open subset U = ψ((−ε, ε) × A) in M. By shrinking A if necessary, we may suppose A is a coordinate patch on Γc and let y 1 , · · · , ym−1 be corresponding coordinate functions. Then {t, y1 , · · · , ym−1} form a set of coordinate functions on U. With respect to these coordinates, and in view of the facts ∂t = X and X ⊥ ∂y i for all i, the Riemannian metric g has the form g = ⟨X, X⟩dt ⊗ dt + hijdyi ⊗ dyj , where hij = g(∂y i , ∂y j ). Since ⟨X, X⟩ = 1 |∇u| 2 , the volume density dVg = 1 |∇u| q det (hij )dtdy1 · · · dym−1 = 1 |∇u| dtdSt . So we conclude that for any ρ ∈ Cc(U), Z M ρf|∇u|dVg = Z U ρfp det Gtdtdy1 · · · dym−1 = Z c+ε c−ε Z Γt∩U ρf dSt dt. Now the conclusion follows from a standard partition of unity argument. □ As a corollary, we get
7LECTURE4:THERIEMANNIANMEASURECorollary 1.6. Suppose the critical values of u form a closed subset? in R, andVol(2t) 0 so that (t,t +e) is free of critical values. Bytaking f = we get, for h e (0,e),Vol(Nt+h) - Vol(2t) =dStaV2ItfollowsdVol(2t) = lim udstdt=rdstJu17d口2.THE LAPLACE-BELTRAMI OPERATORI The divergence of a vector field.Let X be a smooth vector field on M. Take a coordinate patch (U, r',..., m)(which is of course orientable) on M, then the volume elementWg=VGdal ^...darmis locally an n-form on U. Of course one may choose other coordinates on U, thenthe corresponding volume forms are either the same, or differ by a negative sign.Asaresult,thefollowing definition is independent of the choiceof coordinate charts:Definition 2.1. The divergence of X is the function div(X) on M such that(divX)wg = d(c(X)wg).Remark. According to Cartan's magic formula, the definition above is equivalent toCx(wg) = div(X)wg,where Cx is the Lie derivative along the vector field X. This coincides with thegeometric definition of divergence in the case of Rm:the divergence of a vector fieldis the infinitesimal rate of change of the volume element along the vector field.Let's calculatediv(X) locally.Let X =X'a, then(divX)VGdr^.. ^ drm = d (c(X'a,)vGdr" A... ^dam= d(xiVG(-1)-Ida* A...^ dri A... A dam)= a(XVG)dr A... drm2This condition holds if u is a proper function
LECTURE 4: THE RIEMANNIAN MEASURE 7 Corollary 1.6. Suppose the critical values of u form a closed subset2 in R, and Vol(Ωt) 0 so that (t, t + ε) is free of critical values. By taking f = 1 |∇u| we get, for h ∈ (0, ε), Vol(Ωt+h) − Vol(Ωt) = Z t+h t Z Γt 1 |∇u| dSt dt. It follows d dtVol(Ωt) = lim h→0 1 h Z t+h t Z Γt 1 |∇u| dSt dt = Z Γt 1 |∇u| dSt . □ 2. The Laplace-Beltrami operator ¶ The divergence of a vector field. Let X be a smooth vector field on M. Take a coordinate patch (U, x1 , · · · , xm) (which is of course orientable) on M, then the volume element ωg = √ Gdx1 ∧ · · · ∧ dxm is locally an n-form on U. Of course one may choose other coordinates on U, then the corresponding volume forms are either the same, or differ by a negative sign. As a result, the following definition is independent of the choice of coordinate charts: Definition 2.1. The divergence of X is the function div(X) on M such that (divX)ωg = d ι(X)ωg . Remark. According to Cartan’s magic formula, the definition above is equivalent to LX(ωg) = div(X)ωg, where LX is the Lie derivative along the vector field X. This coincides with the geometric definition of divergence in the case of R m: the divergence of a vector field is the infinitesimal rate of change of the volume element along the vector field. Let’s calculate div(X) locally. Let X = Xi∂i , then (divX) √ Gdx1 ∧ · · · ∧ dxm = d ι(X i ∂i) √ Gdx1 ∧ · · · ∧ dxm = d X i X i √ G(−1)i−1 dx1 ∧ · · · ∧ dxci ∧ · · · ∧ dxm = ∂i(X i √ G)dx1 ∧ · · · ∧ dxm, 2This condition holds if u is a proper function
8LECTURE4:THERIEMANNIANMEASUREso we conclude(XVG).div(Xio) =VGWe may replace X by fx to getdiv(fx)= fdivX +(o,f)x= fdivX +g(Vf,X)In other words,Corollary 2.2.For any smooth vector field X EFo(TM) and any smooth functionf ECo(M), one hasdiv(fX)= fdivX +g(Vf,X)Asan application,weproveTheorem 2.3 (TheDivergence theoremI).Let Xbea smooth vector field withcompact support on a Riemannian manifold (M,g), thendiv(X)dVg = 0.Proof. First we assume that X is supported in a local chart (∞,U, V) and X = Xia;with X' e Co(U). Then1div(X)dVg =0(XiVG)dVgVG(xiVGo-1)dal..- drm = 07The general case follows from partition of unity and Corollary 2.2:padiv(X) =div(paX) - g(V(pa), X) =div(paX)and thus( div(X)dVg = /padiv(X)dVg = /div(paX)dV, = 0.1.口IThe Laplace-Beltrami operatorLet (M, g) be a Riemannian manifold.Definition 2.4.Forany smooth function f, we define the Laplacian of f to beAf = -div(Vf)
8 LECTURE 4: THE RIEMANNIAN MEASURE so we conclude div(X i ∂i) = 1 √ G ∂i(X i √ G). We may replace X by fX to get div(fX) = fdivX + (∂if)X i = fdivX + g(∇f, X). In other words, Corollary 2.2. For any smooth vector field X ∈ Γ ∞(TM) and any smooth function f ∈ C ∞(M), one has div(fX) = fdivX + g(∇f, X). As an application, we prove Theorem 2.3 (The Divergence theorem I). Let X be a smooth vector field with compact support on a Riemannian manifold (M, g), then Z M div(X)dVg = 0. Proof. First we assume that X is supported in a local chart (φ, U, V ) and X = Xi∂i with Xi ∈ C ∞ c (U). Then Z M div(X)dVg = Z U 1 √ G ∂i(X i √ G)dVg = Z φ(U) ∂i(X i √ G ◦ φ −1 )dx1 · · · dxm = 0. The general case follows from partition of unity and Corollary 2.2: X α ραdiv(X) = X α div(ραX) − g(∇( X α ρα), X) = X α div(ραX) and thus Z M div(X)dVg = Z M Xραdiv(X)dVg = Z M X α div(ραX)dVg = 0. □ ¶ The Laplace-Beltrami operator. Let (M, g) be a Riemannian manifold. Definition 2.4. For any smooth function f, we define the Laplacian of f to be ∆f = −div(∇f)
9LECTURE4:THERIEMANNIANMEASURELocally,f is given byo,(VGga,f),Af=-div(gia;fo)=VGi.e.Aa(VGg"a,).VGWe shall call the Laplace-Beltrami operator.It is a second order differentialoperator on M, and is the most important differential operator on Riemannianmanifolds. It plays an essential role on the analysis of Riemannian manifolds.Theorem 2.5 (Green's formula I). Suppose f and h are smooth function on M andeither f orh is compactly supported.Then(fAhdVg= /g(Vf,Vh)dVg=hAfdVgProof. We have seendiv(fX) = fdivX + g(Vf, X).It followsdiv(fVh)=-f△h+g(Vf,Vh)口Nowthetheoremfollowsfrom thefact thatfVh is compactlysupported.In particular if M is compact (without boundary), then any smooth function iscompactly supported.Replacing h by h if they are complex-valued, we can rewritethe above formula as(f,△h)12 = (Af,h)L2.In other words, we getCorollary 2.6. If M is compact, then is densely defined symmetric operator onL2(M,g).As another immediate consequence, we see that is a positive operator:Corollary 2.7. If M is compact, then (Af, f)L2 ≥0.Remark. Both the divergence theorem and the Green's formula can be generalizedto the case where M is a compact Riemannian manifold with boundary, i.e. M is.anmdimensional smoothmanifoldwith boundary:M is also a compact subset of an m dimensional Riemannian manifold N.The Riemannian structure on M coincide with that of NSo oM carries(1) an outward normal vector field v(2) an induced Riemannian metric from gn, and thus a volume density dA
LECTURE 4: THE RIEMANNIAN MEASURE 9 Locally, ∆f is given by ∆f = −div(g ij∂if ∂j ) = − 1 √ G ∂i( √ Ggij∂jf), i.e. ∆ = − 1 √ G ∂i( √ Ggij∂j ). We shall call ∆ the Laplace-Beltrami operator. It is a second order differential operator on M, and is the most important differential operator on Riemannian manifolds. It plays an essential role on the analysis of Riemannian manifolds. Theorem 2.5 (Green’s formula I). Suppose f and h are smooth function on M and either f or h is compactly supported. Then Z M f∆hdVg = Z M g(∇f, ∇h)dVg = Z M h∆f dVg. Proof. We have seen div(fX) = fdivX + g(∇f, X). It follows div(f∇h) = −f∆h + g(∇f, ∇h). Now the theorem follows from the fact that f∇h is compactly supported. □ In particular if M is compact (without boundary), then any smooth function is compactly supported. Replacing h by h if they are complex-valued, we can rewrite the above formula as ⟨f, ∆h⟩L2 = ⟨∆f, h⟩L2 . In other words, we get Corollary 2.6. If M is compact, then ∆ is densely defined symmetric operator on L 2 (M, g). As another immediate consequence, we see that ∆ is a positive operator: Corollary 2.7. If M is compact, then ⟨∆f, f⟩L2 ≥ 0. Remark. Both the divergence theorem and the Green’s formula can be generalized to the case where M is a compact Riemannian manifold with boundary, i.e. M is • an m dimensional smooth manifold with boundary • M is also a compact subset of an m dimensional Riemannian manifold N • The Riemannian structure on M coincide with that of N So ∂M carries (1) an outward normal vector field ν (2) an induced Riemannian metric from gN , and thus a volume density dA
10LECTURE4:THERIEMANNIANMEASUREThen for any smooth vector field X on M and any smooth functions f,h on M,div(X)dVg=(Divergence Theorem I) g(X,v)dA,f△hdVg=g(u, Vh)f dA.(Green's formula I)g(Vf, Vh) dVg -Details will be left as an exerciseI Laplacian v.s. isometry.Why the operator is so important in Riemannian geometry? Since differentialoperators are local, it is quite obvious that if β : (M, gM) → (N,gn) is a localisometry, then *(f)=△m(*f).Conversely,Proposition 2.8. A diffeomorphism : M→ N is an isometry between (M,gm)and (N,gn) if and only if it commutes with the Beltrami-Laplace operators, i.e.b*(ANf)=AM(u*f),Vf EC(N).Proof. Obviously if is an isometry, then it commutes with the Beltrami-Laplaceoperators.Conversely,suppose the diffeomorphism commutes with the Beltrami-Laplace operators. Take a chart (, U, V) on M so that (go-1, (U), V) is a charton N.. Denote the coordinates by rl,..., am and y',...,ym respectively. Thenunderthesecoordinates,fory=(r)wehaveW(f 0 b)() = (f 00) 0p-1)(g(r) = (f (00±-1)-1)2(po b-1(y)) =oN f(y)OriOriand thus(ONaN f)ob =OMaM(f ob)On the other hand, we haveON(VGNgNaN f)(()(b*△f)(r) = (△nf)(b(r) =VGN=-(gRoNan f)(()) +..and1o(VGgoy(f o )(z) = -(g%oa)(f o b)(n) +△m(*f)(r) =VGwhere.represents terms that involve only first orderderivatives of f.So bycomparing the coefficients of second order terms, we get g(r)= g((r)), as口desired
10 LECTURE 4: THE RIEMANNIAN MEASURE Then for any smooth vector field X on M and any smooth functions f, h on M, •(Divergence Theorem II) Z M div(X)dVg = Z ∂M g(X, ν)dA, •(Green’s formula II) Z M f∆h dVg = Z M g(∇f, ∇h) dVg − Z ∂M g(ν, ∇h)f dA. Details will be left as an exercise. ¶ Laplacian v.s. isometry. Why the operator ∆ is so important in Riemannian geometry? Since differential operators are local, it is quite obvious that if φ : (M, gM) → (N, gN ) is a local isometry, then ψ ∗ (∆N f) = ∆M(ψ ∗ f). Conversely, Proposition 2.8. A diffeomorphism ψ : M → N is an isometry between (M, gM) and (N, gN ) if and only if it commutes with the Beltrami-Laplace operators, i.e. ψ ∗ (∆N f) = ∆M(ψ ∗ f), ∀f ∈ C ∞(N). Proof. Obviously if φ is an isometry, then it commutes with the Beltrami-Laplace operators. Conversely, suppose the diffeomorphism ψ commutes with the BeltramiLaplace operators. Take a chart (φ, U, V ) on M so that (φ◦ψ −1 , ψ(U), V ) is a chart on N. Denote the coordinates by x 1 , · · · , xm and y 1 , · · · , ym respectively. Then under these coordinates, for y = ψ(x) we have ∂ M i (f ◦ ψ)(x) = ∂((f ◦ ψ) ◦ φ −1 ) ∂xi (φ(x)) = ∂(f ◦ (φ ◦ ψ −1 ) −1 ) ∂xi (φ ◦ ψ −1 (y)) = ∂ N i f(y) and thus (∂ N i ∂ N j f) ◦ ψ = ∂ M i ∂ M j (f ◦ ψ). On the other hand, we have (ψ ∗∆N f)(x) = (∆N f)(ψ(x)) = − 1 √ GN ∂ N i ( p GN g ij N ∂ N j f)(ψ(x)) = −(g ij N ∂ N i ∂ N j f)(ψ(x)) + · · · . and ∆M(ψ ∗ f)(x) = − 1 √ G ∂ M i ( √ Ggij M∂ M j (f ◦ ψ))(x) = −(g ij M∂ M i ∂ M j (f ◦ ψ))(x) + · · · . where · · · represents terms that involve only first order derivatives of f. So by comparing the coefficients of second order terms, we get g ij M(x) = g ij N (ψ(x)), as desired. □