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