《黎曼几何》课程教学资源（英文讲义）Lec09 THE RICCI AND THE SECTIONAL CURVATURE.pdf_大学文库

LECTURE 9: THE RICCI AND THE SECTIONAL CURVATURE 3 Now suppose (M, g) is a Riemannian manifold. Recall that 1 2 g○∧ g is a curvaturelike tensor, such that 1 2 g○∧ g(Xp, Yp, Xp, Yp) = ⟨Xp, Xp⟩⟨Yp, Yp⟩ − ⟨Xp, Yp⟩ 2 , which is nothing else but the square of the area of the parallelogram with sides Xp, Yp. Applying the previous lemma to Rm and 1 2 g○∧ g, we immediately get Proposition 1.4. The quantity K(Xp, Yp) := Rm(Xp, Yp, Xp, Yp) ⟨Xp, Xp⟩⟨Yp, Yp⟩ − ⟨Xp, Yp⟩ 2 depends only on the two dimensional plane Πp = span(Xp, Yp) ⊂ TpM, i.e. it is independent of the choices of basis {Xp, Yp} of Πp. Definition 1.5. We will call K(Πp) = K(Xp, Yp) the sectional curvature of (M, g) at p with respect to the plane Πp. Note that the Ricci curvature that we studied above are closely related to the sectional curvatures: If Xp is a unit vector in TpM, we may extend Xp to an orthonormal basis {e1 = Xp, e2, · · · , em} of TpM. As a result, (1) Ric(Xp)=X i≥2 Rm(ei , e1, ei , e1)=X i≥2 K(ei , e1). In other words, the Ricci curvature Ric(Xp) is “the sum of sectional curvatures” for a specially chosen set of m − 1 pairwise orthogonal planes containing Xp. Example. Here are three basic examples: (1) For the Euclidean space (R m, g0), one has Rm = 0, so K(Πp) ≡ 0 and Ric(Xp) ≡ 0. (2) For the unit sphere (S m, ground), one has Rm = 1 2 g○∧ g, so K(Πp) ≡ 1 and Ric(Xp) ≡ m − 1. One may also prove the conclusion by calculating the Christoffel symbols. (3) For the hyperbolic space (Hm, ghyperbolic), one can prove (exercise) K(Πp) ≡ −1 and Ric(Xp) ≡ −(m − 1). Remark. One may give a conceptional proof of the fact that these three spaces have constant sectional curvature: For (R m, g0), the isometry group E(m) = O(m) ⋉ Rm acts transitively on the set Gr2(TR m) = {Πp | p ∈ R m, Πp ⊂ TpM is a plane}. Since the sectional curvature is invariant under the action of the isometric group, we must have K(Πp) = K(Π′ q ), i.e. the sectional curvature is a constant. The same phenomena occurs for (S m, ground) (with isometry group O(m+1)) and for (Hm, ghyperbolic) (with isometry group O+(m, 1))

4LECTURE9:THERICCIAND THESECTIONALCURVATURERemark. We may compare the sectional curvature K with the curvature operatorR : A2(T,M) → A2(T,M). By definition we have(Rp(Xp ^ Yp), Xp^Yp)K(Xp,Yp) :(Xp, Xp)(Yp, Yp) - (Xp, Yp)2As a consequence, the curvature operator R determines the sectional curvature K.2.BASICPROPERTIESOFSECTIONALCURVATURESI Sectional curvature for low dimensional manifolds.Let (M,g) be a Riemannian manifold. By definition, the sectional curvature Kis NOT a function on M,but instead a function on the Grassmannian bundleGr2(TM) = [(p, II,) I p E M, II, C T,M is 2-dimensional)(1) If M has dimension m = 1, obviously R = 0 and it makes no sense to talkabout sectional curvature (so again as we have seen in Lecture 3, essentially there isnoRiemanniangeometryindimensionalone)(2) If M has dimension m = 2, i.e. is a surface, then one can regard the sectionalcurvature K as a function defined on M in the natural way. Moreover, inview of the equation (1), for any direction X, we have Ric(X,) = K(p). Inother words, for surfaces the sectional curvature, the Ricci curvature and thescalar curvature are all the same.One can show that in this case K is reallythe Gauss curvature in undergraduate differential geometry course.(3) If M has dimension m ≥ 3, by using the exponential map that we willlearn later, for each 2-dimensional plane IIp E T,M, locally one gets a 2-dimensional submanifold Spnear p in M whose tangent space atp is Ip,and the sectional curvature K(II,) is nothing else but the Gauss curvatureof S, (endowed with the subspace Riemannian metric) at p. Thus the sec-tional curvature is a natural generalization of the Gauss curvature to higherdimensional Riemannian manifolds.(4) In general the sectional curvatures encodes more information than the Riccicurvatures.However,ifMhasdimensionm=3,thenaccordingtotheequation (1), for an orthonormal basis e1, e2, e3,Ric(ei) = K(e1,e2) + K(e1,e3),Ric(e2) = K(ei, e2) + K(e2, e3),Ric(e3) = K(e1, e3) +K(e2, e3).As a result, in this case the Ricci curvatures determine the sectional curva-tures: For each plane IIp, one just start with an orthonormal basis [ei,e2]of Ip, extend it to an orthonormal basis [ei, e2, es) of T,M and then solvethe above system of equations to get2K(Ilp) = Ric(e1) + Ric(e2) -Ric(e3)

4 LECTURE 9: THE RICCI AND THE SECTIONAL CURVATURE Remark. We may compare the sectional curvature K with the curvature operator Rp : Λ2 (TpM) → Λ 2 (TpM). By definition we have K(Xp, Yp) = ⟨Rp(Xp ∧ Yp), Xp ∧ Yp⟩ ⟨Xp, Xp⟩⟨Yp, Yp⟩ − ⟨Xp, Yp⟩ 2 . As a consequence, the curvature operator R determines the sectional curvature K. 2. Basic properties of sectional curvatures ¶ Sectional curvature for low dimensional manifolds. Let (M, g) be a Riemannian manifold. By definition, the sectional curvature K is NOT a function on M, but instead a function on the Grassmannian bundle Gr2(TM) = {(p, Πp) | p ∈ M, Πp ⊂ TpM is 2-dimensional}. (1) If M has dimension m = 1, obviously R ≡ 0 and it makes no sense to talk about sectional curvature (so again as we have seen in Lecture 3, essentially there is no Riemannian geometry in dimensional one). (2) If M has dimension m = 2, i.e. is a surface, then one can regard the sectional curvature K as a function defined on M in the natural way. Moreover, in view of the equation (1), for any direction Xp we have Ric(Xp) = K(p). In other words, for surfaces the sectional curvature, the Ricci curvature and the scalar curvature are all the same. One can show that in this case K is really the Gauss curvature in undergraduate differential geometry course. (3) If M has dimension m ≥ 3, by using the exponential map that we will learn later, for each 2-dimensional plane Πp ∈ TpM, ✿✿✿✿✿✿✿ locally one gets a 2- dimensional submanifold Sp near p in M whose tangent space at p is Πp, and the sectional curvature K(Πp) is nothing else but the Gauss curvature of Sp (endowed with the subspace Riemannian metric) at p. Thus the sectional curvature is a natural generalization of the Gauss curvature to higher dimensional Riemannian manifolds. (4) In general the sectional curvatures encodes more information than the Ricci curvatures. However, if M has dimension m = 3, then according to the equation (1), for an orthonormal basis e1, e2, e3, Ric(e1) = K(e1, e2) + K(e1, e3), Ric(e2) = K(e1, e2) + K(e2, e3), Ric(e3) = K(e1, e3) + K(e2, e3). As a result, in this case the Ricci curvatures determine the sectional curvatures: For each plane Πp, one just start with an orthonormal basis {e1, e2} of Πp, extend it to an orthonormal basis {e1, e2, e3} of TpM and then solve the above system of equations to get 2K(Πp) = Ric(e1) + Ric(e2) − Ric(e3)

LECTURE 9:THERICCI AND THESECTIONALCURVATURE5I Riemannian manifolds with curvature bounds.Unlikealgebraicquantitieslikecurvaturetensors,thesectional/Ricci/scalarcur-vatures are real-valued functions. Since we may compare real nubmers, we can defineDefinition 2.1. Let (M,g) be a Riemannian manifold. We say (M, g) has(1)constant sectional curvaturecifK(1,)=cforall pand all planes ,CT,M.(2) constant Ricci curvature c if Ric(X,) = c for all p and all vectors X, E SpM.(3) positive sectional curvature if K(IIp) >0 for all p and all planes II, C T,M.Similarly wemay define.(M,g)has negative/nonpositive/nonnegative sectional curvature if K(1I,)isnegative/nonpositive/nonnegative for all p and all planes II, C T,M.. (M, g) has positive/negative/nonpositive/nonnegative Ricci curvature if Ric(Xp)ispositive/negative/nonpositive/nonnegativeforallpandall XnESnM..(M,g) has sectional curvatureK≥c or K≤ c if K(1Ip)≥ c or K(1lp)≤cfor all p and all planes IIp T,M..(M,g) has Ricci curvature Ric≥ c orRic≤ cif Ric(X,)≥c or Ric(X,)≤ cforallpand all vectorsX,S,M.More generally, given two Riemannian metrics g1 and g2 on any smooth manifoldM, with sectional curvature functions Kgr, Kg, and Ricci curvature functions Ricgn,Ricga, we may compare Kg, and Kg, as functions on Gr2(TM), and compare Ricgiand Ricga as functions on SM.Erample.We can write down the change of sectional curvature under scaling of themetric: if we scale a Riemannian metric g to Ag, where A is a positive constant, then(1)by the Koszul formula, the Levi-Civita connection xY remains unchanged.(2) it follows that the (1,3)-curvature tensor R remains unchanged,(3) as a result, the Riemann curvature tensor is changed to Rmxg = ARmg,(4) and thus the Ricci curvature tensor remains unchanged: Rcag = Rcg,(5)but the two sectional curvatures are related by Kxg =^-1Kg,(6) and it follows that Ricag =^-1Ricg (no conflict with (4) since unit vectors changed)(7) and thus Sxg = >-1Sg.In particular, for each c one gets a Riemannian manifold with constant sectionalcurvature c, namely the space (sm, Iground) if c > 0, and (Hm, ghuperbolic) if c < 0.Manifolds with curvature bounds will be one of the major themes of this course.Remark.Since the curvature operator R,is a symmetric operator on a real vectorspace, it has real eigenvalues.Definition 2.2. We say (M, g) is a Riemannian manifold with posi-tive curvature operator if all eigenvalues of Rp are positive

LECTURE 9: THE RICCI AND THE SECTIONAL CURVATURE 5 ¶ Riemannian manifolds with curvature bounds. Unlike algebraic quantities like curvature tensors, the sectional/Ricci/scalar curvatures are ✿✿✿✿✿✿✿✿✿✿✿ real-valued functions. Since we may compare real nubmers, we can define Definition 2.1. Let (M, g) be a Riemannian manifold. We say (M, g) has (1) constant sectional curvature c if K(Πp) = c for all p and all planes Πp ⊂ TpM. (2) constant Ricci curvature c if Ric(Xp) = c for all p and all vectors Xp ∈ SpM. (3) positive sectional curvature if K(Πp) > 0 for all p and all planes Πp ⊂ TpM. Similarly we may define • (M, g) has negative/nonpositive/nonnegative sectional curvature if K(Πp) is negative/nonpositive/nonnegative for all p and all planes Πp ⊂ TpM. • (M, g) has positive/negative/nonpositive/nonnegative Ricci curvature if Ric(Xp) is positive/negative/nonpositive/nonnegative for all p and all Xp ∈ SpM. • (M, g) has sectional curvature K ≥ c or K ≤ c if K(Πp) ≥ c or K(Πp) ≤ c for all p and all planes Πp ⊂ TpM. • (M, g) has Ricci curvature Ric ≥ c or Ric ≤ c if Ric(Xp) ≥ c or Ric(Xp) ≤ c for all p and all vectors Xp ∈ SpM. More generally, given two Riemannian metrics g1 and g2 on any smooth manifold M, with sectional curvature functions Kg1 , Kg2 and Ricci curvature functions Ricg1 , Ricg2 , we may compare Kg1 and Kg2 as functions on Gr2(TM), and compare Ricg1 and Ricg2 as functions on SM. Example. We can write down the change of sectional curvature under scaling of the metric: if we scale a Riemannian metric g to λg, where λ is a positive constant, then (1) by the Koszul formula, the Levi-Civita connection ∇XY remains unchanged, (2) it follows that the (1,3)-curvature tensor R remains unchanged, (3) as a result, the Riemann curvature tensor is changed to Rmλg = λRmg, (4) and thus the Ricci curvature tensor remains unchanged: Rcλg = Rcg, (5) but the two sectional curvatures are related by Kλg = λ −1Kg, (6) and it follows that Ricλg = λ −1Ricg (no conflict with (4) since unit vectors changed) (7) and thus Sλg = λ −1Sg. In particular, for each c one gets a Riemannian manifold with constant sectional curvature c, namely the space (S m, 1 c ground) if c > 0, and (Hm, 1 −c ghyperbolic) if c < 0. Manifolds with curvature bounds will be one of the major themes of this course. Remark. Since the curvature operator Rp is a symmetric operator on a real vector space, it has real eigenvalues. Definition 2.2. We say (M, g) is a Riemannian manifold with positive curvature operator if all eigenvalues of Rp are positive

6 LECTURE 9: THE RICCI AND THE SECTIONAL CURVATURE Obviously if (M, g) has positive curvature operator, then it has positive sectional curvature. However, the converse is not true: • It was proven by C. Bohm and B. Wilking in 2008 that manifolds with positive curvature operators are space forms, i.e. are complete Riemannian manifolds with constant sectional curvature. • On the other hands, there exist Riemannian manifolds with positive sectional curvature which are not constant (e.g. the complex projective space CPm endowed with the Fubini-Study metric). So for such Riemannian manifolds the curvature operator is not positive. The secret is: the space Λ2 (TpM) is a vector space that contains elements of the form u1 ∧v1 + u2 ∧ v2 which do not correspond to any 2-dimensional plane in TpM. After all, the Grassmannian Gr2(TpM) is a smooth manifold of dimension 2(m − 2), while the space of bi-vectors Λ2 (TpM) has dimension m 2 . ¶ Riemannian manifolds with isotropic sectional curvature at a point. Finally study the following question: At a given point, when will the sectional curvature be independent of the choice of Πp ⊂ TpM? Proposition 2.3. Let (M, g) be a Riemannian manifold and p ∈ M. The following are equivalent: (1) K(Πp) = c for all Πp ⊂ TpM. (2) Rmp = c 2 gp○∧ gp. (3) Rp(Xp, Yp)Zp = c(⟨Yp, Zp⟩Xp − ⟨Xp, Zp⟩Yp) for any Xp, Yp, Zp ∈ TpM. (4) Rp = cId on Λ 2TpM. (5) The Weyl curvature tensor Wp = 0 and Ricci curvature tensor Rcp = (m − 1)cgp. Proof. (1) ⇐⇒ (2) According to Lemma 1.2, if T is a curvature-like tensor, then T ≡ 0 ⇐⇒ T(X, Y, X, Y ) = 0, ∀X, Y. Apply this to the curvature-like tensor T = Rmp − c 2 gp○∧ gp, we see Rmp = c 2 gp○∧ gp ⇐⇒ K(Πp) = c, ∀ Πp ⊂ TpM. (2) ⇐⇒ (3) and (4) =⇒ (1) are obvious. (2) =⇒ (4) Take an orthonormal basis {ei} of TpM, then {ei ∧ ej | i < j} is a basis of Λ2TpM. On this basis, ⟨Rp(ei ∧ ej ), ek ∧ el⟩ = Rmp(ei , ej , ek, el) = cδikδjl, where in the last step we used the fact i < j, k < l. As a result, we see Rp(ei ∧ ej ) = cei ∧ ej , ∀i < j. In other words, Rp = cId for all ei ∧ ej . Since Rp is linear, Rp = cId on Λ2TpM