About the Injectivity Radius and the Ricci Tensor of a Complete Riemannian Manifold
aa r X i v : . [ m a t h . DG ] D ec ABOUT THE INJECTIVITY RADIUS AND THE RICCITENSOR OF A COMPLETE RIEMANNIAN MANIFOLD
S´ergio L. Silva
Abstract.
In this paper we obtain a simple upper bound for the infimum of theRicci curvatures of a complete Riemannian manifold M n with nonzero injectivityradius i ( M ) depending only on of the i ( M ). In case of rigidity the Riemannianmanifold must be an n -dimensional Euclidean sphere(Euclidean space) conform theinjectivity radius be finite(infinite). Furthermore with the additional assumptionthat the second derivative of the Ricci tensor of M n is null we prove that the sameupper bound for the infimum of the Ricci curvatures holds for the supremum of theRicci curvatures and M n has, in fact, parallel Ricci tensor.
1. Introduction
Let M n be a differentiable n -dimensional complete Riemannian manifold withmetric h , i and Levi-Civita connection ∇ . The curvature tensor R of M n is definedby R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z for all differentiable vector fields X, Y and Z on M n .We denote by T p M the tangent space of M n at p , exp p the exponential map of M n at p , d the intrinsic distance function on M n an by C ( p ) the cut locus of p .The injectivity radius of M n is given by i ( M ) = inf p ∈ M d ( p, C ( p )) . The
Ricci Tensor of M n , denoted by Ric , is the differentiable symmetric 2-form
Ric p ( v, w ) = 1 n − n X ı =1 h R p ( v, e ı ) e ı , w i for all p ∈ T p M and v, w ∈ T p M , being e , . . . , e n an orthonormal basis of T p M .Given p ∈ M n and a unitary vector v tangent to M n at p , the number Ric p ( v, v )is called the Ricci curvature of M n at p in the direction v .For a differentiable k -form ω on M n its covariant derivative is the ( k + 1)-form ∇ ω given by ∇ ω ( X , X , . . . , X k +1 ) = X ( ω ( X , . . . , X k +1 )) − k +1 X ı =2 ω (cid:0) X , . . . , ∇ X X ı , . . . , X k +1 (cid:1) Mathematics Subject Classification.
Primary ; Secondary.
Key words and phrases. complete, Riemannian manifold, ricci tensor, infectivity radius. S´ERGIO L. SILVA for all vector fields X , X , . . . , X k +1 on M n . For m ≥
2, we put ∇ m ω = ∇ (cid:0) ∇ m − ω (cid:1) . We say that ω is parallel on M n when ∇ ω ≡ T M denote the unitary tangent bundle of M n , that is, T M = { ( q, w ) : q ∈ M n , w ∈ T q M, || v || = 1 } . Our first result is a simple upper bound for the infimum of the Ricci curvature ofa complete Riemannian manifold with nonzero injectivity radius as a function on T M . If i ( M ) is infinite follows immediately from Myers’ and Bonnet’s Theoremsthat inf ( q,w ) ∈ T M Ric q ( w, w ) ≤
0. Note that if equality holds then M n is isometric tothe Euclidean space R n by a splitting theorem(see Theorem 2 in [C-G]) since anygeodesic is a line. When i ( M ) is finite we obtain the following result Theorem 1.1.
Let M n be a complete Riemannian manifold with < i ( M ) < ∞ .Then, inf ( q,w ) ∈ T M Ric q ( w, w ) ≤ (cid:20) πi ( M ) (cid:21) . If equality holds then M n is isometric to the sphere of diameter i ( M ) . Our next result establishes that if a complete Riemannian manifold satisfies ∇ Ric ≡ M n has, in fact, parallel Ricci tensor and the same upper boundfor the infimum of the Ricci curvatures also holds for the supremum of the Riccicurvature. More precisely, we have the following result Theorem 1.2.
Let M n be a complete Riemannian manifold with ∇ Ric ≡ . Thenthe following hold:(i) If i ( M ) = 0 then the Ricci tensor of M n is parallel. Furthermore, if i ( M ) = ∞ then the Ricci curvature is non-positive on M n and if < i ( M ) < ∞ then sup ( q,w ) ∈ T M Ric q ( w, w ) ≤ h πi ( M ) i .(ii) If n = 2 ( M is a surface) then M has constant Gaussian curvature.
2. Proofs of Theorem 1.1 and Theorem 1.2
Proof of Theorem 1.1.
Given ( p, v ) ∈ T M and a real number t o such that 0 < t o We have ∂f i ∂t ( t, 0) = γ ′ ( t ), ∂f i ∂s ( t, 0) = V i ( t ) and Dds ∂f i ∂s ( t, s ) = 0. Consequently,12 E ′′ i (0) = − Z t o sin (cid:18) πtt o (cid:19) h R ( γ ′ ( t ) , e i ( t )) e i ( t ) , γ ′ ( t ) i dt (2.3) + Z t o (cid:18) πt o (cid:19) cos (cid:18) πtt o (cid:19) dt. and 12 n X ı =2 E ′′ i (0) = − ( n − Z t o sin (cid:18) πtt o (cid:19) Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) dt (2.4) + ( n − Z t o (cid:18) πt o (cid:19) cos (cid:18) πtt o (cid:19) dt. Due to (2.1) we can write0 ≤ − Z t o sin (cid:18) πtt o (cid:19) Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) dt + Z t o (cid:18) πt o (cid:19) cos (cid:18) πtt o (cid:19) dt ≤ − inf ( q,w ) ∈ T M Ric q ( w, w ) Z t o sin (cid:18) πtt o (cid:19) dt + Z t o (cid:18) πt o (cid:19) cos (cid:18) πtt o (cid:19) dt = t o "(cid:18) πt o (cid:19) − inf ( q,w ) ∈ T M Ric q ( w, w ) . The above inequality implies that inf ( q,w ) ∈ T M Ric q ( w, w ) ≤ (cid:16) πt o (cid:17) for all t o such that0 < t o < d ( p, C ( p )). If d ( p, C ( p )) is infinite for some p then inf ( q,w ) ∈ T M Ric q ( w, w ) ≤ d ( p, C ( p )) finitefor all p ∈ M . In this case,inf ( q,w ) ∈ T M Ric q ( w, w ) ≤ (cid:18) πd ( p, C ( p )) (cid:19) . Since i ( M ) = inf p ∈ M d ( p, C ( p )) and i ( M ) is finite and nonzero the inequality in Theo-rem 1.1 follows. If equality holds then Ric q ( w, w ) ≥ (cid:16) πi ( M ) (cid:17) for all ( q, w ) ∈ T M .So by Myers’ and Bonnet’s Theorems M n is compact and has diameter less thanor equal to i ( M ). Consequently, the diameter of M n is i ( M ). Now that M n isisometric to the sphere of diameter i ( M ) follows from Theorem 3.1 in [SC](see also[KS]). (cid:3) Before proving Theorem 1.2 we prove the following proposition Proposition 2.1. Let M n be a complete Riemannian manifold with ∇ Ric ≡ .Then for all p ∈ M n , all unitary vector v in T p M and all t o such that < t o Proof. Since ∇ Ric ≡ 0, for the normalized geodesic γ such that γ (0) = p and γ ′ (0) = v , we have(2.5) Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) = a γ t + b γ , t ∈ R , with a γ = ∇ Ric p ( v, v, v ) and b γ = Ric p ( v, v ). As in proof of Theorem 1.1 if weconsider the variations f i ( t, s ), i = 2 , . . . , n , using (2.1), (2.4) and (2.5), we deducethat(2.6) 0 ≤ − Z t o sin (cid:18) πtt o (cid:19) ( a γ t + b γ ) dt + Z t o (cid:18) πt o (cid:19) cos (cid:18) πtt o (cid:19) dt. Thus,(2.7) 0 ≤ − a γ t o − b γ t o π t o , that is,(2.8) 0 ≤ − a γ t o − b γ t o + 2 π t o . If we use the geodesic β such that β (0) = p and β ′ (0) = − v , proceeding analogousto the above, give us(2.9) 0 ≤ a γ t o − b γ t o + 2 π t o . As a consequence of (2.8) and (2.9) we have0 ≤ − b γ t o + 4 π t o . and(2.10) Ric p ( v, v ) = b γ ≤ (cid:18) πt o (cid:19) . (cid:3) Corollary 2.2. Let M n be a complete Riemannian manifold with ∇ Ric ≡ . If γ is a normalized ray satisfying γ (0) = p and γ ′ (0) = v then a γ = ∇ Ric p ( v, v, v ) ≤ .Case a γ = 0 , we must have b γ = Ric p ( v, v ) ≤ . Equivalently, Ric γ ( γ ′ , γ ′ ) is eitherdecreasing or constant non-positive when γ is a ray.Proof. Since γ is a ray it is minimizing from 0 to t o for all t o > 0. Then theinequality (2.8) holds for all t o > 0. Consequently, a γ = ∇ Ric p ( v, v, v ) ≤ 0. If a γ < Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) = a γ t + b γ , t ∈ R , is decreasing. Case a γ = 0, wehave Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) = b γ for all t ∈ R and is clear from (2.8) that b γ ≤ (cid:3) NJECTIVITY RADIUS AND THE RICCI TENSOR 5 Corollary 2.3. Let M n be a complete Riemannian manifold with ∇ Ric ≡ . If γ is a normalized line satisfying γ (0) = p and γ ′ (0) = v then ∇ Ric p ( v, v, v ) = 0 and Ric p ( v, v ) ≤ . Equivalently, Ric γ ( γ ′ , γ ′ ) is constant non-positive when γ is aline.Proof. Since γ is a line the inequalities (2.8) and (2.9) hold for all t o > 0. Thus, a γ = ∇ Ric p ( v, v, v ) = 0 and Ric γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) = b γ for all t ∈ R with b γ = Ric p ( v, v ) ≤ (cid:3) Corollary 2.4. Let M n be a complete Riemannian manifold with ∇ Ric ≡ . If d ( p, C ( p )) = ∞ then Ric p ( v, v ) ≤ for all unitary vector v ∈ T p M .Proof. Since d ( p, C ( p )) = ∞ , by Proposition 2.1, the inequality Ric p ( v, v ) ≤ (cid:16) πt o (cid:17) holds for all unitary vector v ∈ T p M and all t o > 0. Consequently, Ric p ( v, v ) ≤ v ∈ T p M . (cid:3) Proof of Theorem 1.2. (i) If i ( M ) = ∞ then d ( p, C ( p )) = ∞ for all p ∈ M n and the Corollary 2.4gives Ric p ( v, v ) ≤ p ∈ M n and all unitary vector v ∈ T p M . If 0
0. Being p arbitrary, we conclude that M n has parallel Ricci tensor. (cid:3) (ii) If dim M = 2 then the Ricci curvature Ric p ( v, v ), for all p ∈ M n and allunitary vector v ∈ T p M , is the Gaussian curvature at p denoted by K ( p ). Sofor a normalized geodesic γ we have K γ ( t ) = K ( γ ( t )) = a γ t + b γ , t ∈ R , where a γ and b γ are real constants depending on γ . We affirm that K γ is constant forany geodesic γ . In fact, suppose on the contrary that a γ = 0 for some geodesic γ . Changing γ by β such that β ( t ) = γ ( − t ) if necessary we can assume that a γ > 0, that is, K γ is increase. Observe that the geodesic γ is not contained in anycompact set of M n since K γ is unbounded. Consider t o > K γ ( t o ) > dK γ ( t o ) ( γ ′ ( t o )) = a γ > dK γ ( t o ) ( − γ ′ ( t o )) = − a γ < 0, there existsan unitary vector w ∈ T γ ( t o ) M such that dK γ ( t o ) ( w ) = 0. Let θ be the geodesicwith θ (0) = γ ( t o ) and θ ′ (0) = w . It holds that K θ ( t ) = K ( θ ( t )) = a θ t + b θ with a θ = K ′ θ (0) = dK γ ( t o ) ( w ) = 0. Consequently, K θ is constant. Take a sequence ofpositive real numbers t n such that t n → ∞ and that p n = γ ( t o + t n ) be divergent.Fixed a point q = θ ( s ), consider a normalized minimal geodesic γ n from q to p n .Observe that b θ = K θ ( s ) = K ( q ) = K θ (0) = K ( γ ( t o )) = K γ ( t o ) > K isconstant over θ . Supposing γ n ( t ) = exp q tu n and changing u n by a subsequence if S´ERGIO L. SILVA necessary we may assume u n → u . The geodesic α ( t ) = exp q tu , t ∈ R , is a ray.Then dK q ( u ) = a α ≤ n , K γ n is increasing since is anaffine function and K γ n (0) = K ( q ) = K ( γ ( t o )) < K ( γ ( t o + t n )) = K γ n ( d ( q, γ ( t o + t n )) . Recall that K γ is increasing. Thus K ′ γ n ( t ) = K ′ γ n (0) = dK q ( u n ) > 0. Taking thelimit we deduce that dK q ( u ) = lim n →∞ dK q ( u n ) ≥ 0. Then, dK q ( u ) = 0 and K α isconstant non-positive along α by Corollary 2.2. But K α (0) = K ( q ) > K γ is constant for any normalized geodesic γ . Now followsthat M has constant Gaussian curvature since it is complete. (cid:3) References [C-E] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland,Amsterdam, 1975.[C-G] J. Cheeger and Detlef Gromoll, The Splitting Theorem for Manifolds of Nonnegative RicciCurvature , J. Differential Geom. (1971), 119–128.[KS] K. Shiohama, A Sphere Theorem for Manifolds of Positive Ricci Curvature , Trans. A.M.S. (1983), 811–819[MC] M.P. do Carmo, Riemannian Geometry, Birkh¨auser, Boston-Basel-Berlin. Translated byFrancis Flaherty, 1992.[SC] S.Y. Cheng, Eigenvalues Comparison Theorems and Its Geometric Applications , Math. Z. (1975), 289–297 Departamento de Estruturas Matem´aticas, IME, Universidade do Estado do Riode Janeiro, 20550-013, Rio de Janeiro, Brazil. E-mail address ::