arXiv:1710.00276v3 [math.DG] 25 Nov 2017 e ( Let Introduction 1 p Ricci manifold, motion. Einstein Brownian manifold, semigroup, curvature Constant 58J50. Keywords: 58J32, Classification: subject AMS udeo agn etr of vectors tangent of bundle and x respectively. oeconstant some ewl denote will We . ∗ dniyn osatCraueManifolds, Curvature Constant Identifying When upre npr yNSC(1736 11431014). (11771326, NNSFC by part in Supported S M, eie o h etsmgopo isenadRciparalle Ricci and Einstein on semigroup heat the e Moreover, for derived inequalities. semigroup and Ricc formulas the and analytic manifolds stoch Combining Einstein from manifolds, developed curvature curvature. semigroup constant heat Ricci the of the formulas for Hessian formula derivative a ec eateto ahmtc,SasaUiest,SnltnPark, Singleton University, Swansea Mathematics, of Department isenMnfls n ic Parallel Ricci and Manifolds, Einstein g eetbihvrainlfrua o ic pe n lower and upper Ricci for formulas variational establish We eoeteReana uvtr esr ic uvtr n se and curvature Ricci tensor, curvature Riemannian the denote ea be ) M a osatRcicraue i.e. curvature, Ricci constant has etro ple ahmtc,TajnUiest,Tajn307,C 300072, Tianjin University, Tianjin Mathematics, Applied of Center K d h erci aumslto fEnti edeutos(nparticula (in equations field Einstein of solution vacuum a is metric the , dmninlcmlt imninmnfl.Let manifold. Riemannian complete -dimensional h ,v u, i = agytueuc,[email protected] [email protected], g M ( ,v u, hr o every for where , and ) oebr7 2018 7, November Manifolds e gY Wang Feng-Yu | Abstract u | = 1 p R h x ic ,u u, ∈ = M i K for , ∗ g T x ,v u, sml eoeby denote (simply pii esa siae are estimates Hessian xplicit M aallmnflsb using by manifolds parallel i manifolds. l si nlss eidentify we analysis, astic stetnetsaea point at space tangent the is A P,Uie Kingdom United 8PP, SA2 ∈ ewt eiaieand derivative with se T x M T ons swl as well as bounds, ,x M, rle aiod heat manifold, arallel = ∈ hina ∪ toa curvature ctional x M. ∈ M R T ic Let x M = R K ethe be , for ) R ic r for d = 4 in general relativity), and M is called an Einstein manifold. In differential ge- ometry, a basic problem is to characterize topological and geometry properties of Einstein manifolds. For instance, according to Thorpe [16] and Hitchin [17], if a four-dimensional compact manifold admits an Einstein metric, then 3 χ(M) τ(M) , ≥ 2| | where χ is the Euler characterization and τ is the signature, and the equality holds if and only if M is a flat torus, a Calabi-Yau manifold, or a quotient thereof. Recently, Brandle [6] showed that Einstein manifolds with positive isotropy curvature are space forms, while with nonnegative isotropy curvature are locally symmetric, and an ongoing investigation is to classify Einstein manifolds of both positive and negative sectional curvatures. Since the curvature tensor is determined by the sectional curvature, M is called a constant curvature manifold if ec = k for some constant k R. Complete simply connected constant S ∈ curvature manifolds are called space forms, which are classified by hyperbolic space (negative constant sectional curvature), Euclidean space (zero sectional curvature) and unit sphere (positive sectional curvature) respectively. All other connected complete constant curvature manifolds are quotients of space forms by some group of isometries. Constant curvature manifolds are Einstein but not vice versa. A slightly larger class of manifolds are Ricci parallel manifolds, where the Ricci curvature is constant under parallel transports; that is, ic = 0 where is the Levi-Civita connec- tion. An Einstein manifold is Ricci parallel but∇R the inverse is not∇ true. When the manifold is simply connected and indecomposable (i.e. does not split as non-trivial Riemannian prod- ucts), these two properties are equivalent, see e.g. [11, Chapter XI]. In this paper, we aim to identify the above three classes of manifolds by using integral formulas with respect to the volume measure and derivative inequalities of the heat semi- group. To state the main results, we introduce some notations where most are standard in the literature. For f,g C2(M) and x M, consider the Hilbert-Schmidt inner product of the Hessian ∈ ∈ tensors Hessf and Hessg:
d Hess , Hess (x)= Hess (Φi, Φj)Hess (Φi, Φj), h f giHS f g i,jX=1 where Φ = (Φi) O (M), the space of orthonormal bases of T M. Then the Hilbert- 1≤i≤d ∈ x x Schmidt norm of Hessf reads
Hessf HS = Hessf , Hessf HS . k k qh i For a symmetric 2-tensor T and a constant K, we write K if T ≥ (u,u) K u 2, u TM. T ≥ | | ∈ Similarly, K means (u,u) K u 2,u TM. Let # : T M T M be defined by T ≤ T ≤ | | ∈ T → #(u), v = (u, v), u,v T M, x M. hT i T ∈ x ∈ 2 Then # is a symmetric map, i.e. #u, v = #v,u for u, v T M, x M. Let T hT i hT i ∈ x ∈ (x) = sup #(u) : u T M, u 1 , x M. kT k |T | ∈ x | | ≤ ∈ 1 A C map Q : T M T M with QTxM TxM for x M is called constant, if (Qv)(x)=0 holds for any x M→ and vector field v⊂with v(x)∈ = 0. So, M is Ricci parallel∇ if only if ∈ ∇ ic# is a constant map. R For any symmetric 2-tensor , define T d (1.1) ( )(v , v ) := tr ( , v )v , #( ) = (Φi, v )v , #(Φi) , RT 1 2 R · 2 1 T · R 2 1 T Xi=1 where v , v T M, x M, Φ=(Φi) O (M). Since T is symmetric, so is T . Let 1 2 ∈ x ∈ 1≤i≤d ∈ x R (x) = sup (x): is a symmetric 2-tensor, (x) 1 , kRk kRT k T kT k ≤ ∞ = sup (x). kRk x∈M kRk
For a smooth tensor , consider the Bochner Laplacian T ∆ := tr . T ∇·∇·T t 1 2 ∆ 2 Then 2 ∆ generates a contraction semigroup Pt = e in the L space of tensors, see [13, Theorems 2.4 and 3.7] for details. In Subsection 3.1, we will prove a probabilistic formula of P , which is a smooth tensor when is smooth with compact support. Precisely, for tT T any x M and Φ O (M), let Φ (x) be the horizontal Brownian motion starting at Φ, ∈ ∈ x t and let Xt(x) := πΦt(x) be the Brownian motion starting at x, see (3.3) and (3.5) below for details. Then = Φ (x)Φ−1 : T M T M is called the stochastic parallel transport kt t x → Xt(x) along the Brownian path. Both Xt(x) and t do not depend on the choice of the initial value Φ O (M). When the manifold is stochasticallyk complete (i.e. the Brownian motion ∈ x is non-explosive), for a bounded n-tensor we have T (P )(v , , v )= E ( v , , v ) , v , , v T M. tT 1 ··· n T kt 1 ··· kt n 1 ··· n ∈ x 2 In the following, we will take this regular (rather than L ) version of the heat semigroup Pt.
Finally, For v TxM, let Wt(v) TXt(x)M solve the following covariant differential equation ∈ ∈ d 1 (1.2) Φ (x)−1W (v)= Φ−1(x) ic#(W (v)), W (v)= v. dt t t −2 t R t 0
Wt is called the damped stochastic parallel transport. When ic K for some constant K − 2 t R ≥ K R, we have Wt(v) e v , t 0, v TxM. ∈In Section 2| and Sections| ≤ 6-8,| | we≥ will present∈ a number of identifications of constant curvature manifolds, Einstein manifolds, and Ricci parallel manifolds. In particular, the following assertions are direct consequences of Theorems 2.1, 6.1, 7.1 and 8.1 below.
3 (A) Constant curvature. Let k R. Each of the following assertions is equivalent to ec = k: ∈ S (A ) For any t 0 and f C∞(M), Hess = e−dktP Hess + 1 (1 e−dkt)(P ∆f)g. 1 ≥ ∈ 0 Ptf t f d − t (A ) For any f C∞(M), Hess ∆Hess =2k (∆f)g d Hess . 2 ∈ 0 ∆f − f − f (A ) For any f C∞(M), 3 ∈ 0 1 ∆ Hess 2 Hess , Hess Hess 2 =2k d Hess 2 (∆f)2 , 2 k f kHS −h ∆f f iHS − k∇ f kHS k f kHS − 2 d 2 i where Hess := i Hess , Φ=(Φ ) O (M). k∇ f kHS i=1 k∇Φ f kHS 1≤i≤d ∈ x P ∞ (A4) For any x M,u TxM and f C0 (M) with Hessf (x)= u u (i.e. Hessf (v1, v2)= u, v u, v∈ , v , v ∈ T M), ∈ ⊗ h 1ih 2i 1 2 ∈ x Hess ∆Hess (v, v)=2k u 2 v 2 u, v 2 , v T M. ∆f − f | | | | −h i ∈ x According to the Bochner-Weitzenb¨ock formula, for any constant K R, ic = K is equivalent to each of the following formulas: ∈ R
t P f = e− 2 K P f, f C∞(M), t 0, ∇ t t∇ ∈ 0 ≥ ∆ f ∆f = K f, f C∞(M), ∇ −∇ ∇ ∈ 0 1 ∆ f 2 ∆f, f Hess 2 = K f 2, f C∞(M). 2 |∇ | − h∇ ∇ i−k f kHS |∇ | ∈ 0
So, (A1)-(A3) can be regarded as the corresponding formulas for ec = k. Below, we present some other identifications of Einstein manifolds. S
(B) Einstein manifolds. Let µ be the volume measure, and denote µ(f) = M fdµ for f L1(µ). M is Einstein if and only if R ∈ 2 2 µ f, g µ (∆f) Hessf (1.3) h∇ ∇ i −k kHS = µ f 2 µ (∆f)(∆g) Hess ,Hess , f,g C∞(M). |∇ | −h f giHS ∈ 0 Moreover, for any K R, ic = K is equivalent to each of the following assertions: ∈ R (B ) < , and for any t 0, f C∞(M), 1 kRk∞ ∞ ≥ ∈ 0 t −Kt −Ks HessPtf = e PtHessf + e Ps( HessPt−sf )ds. Z0 R
(B ) For any f C∞(M), 2 ∈ 0 1 Hess ∆Hess =( Hess ) KHess . 2 ∆f − f R f − f 4 (B ) For any f C∞(M), 3 ∈ 0 µ (∆f)2 Hess 2 = Kµ f 2 . −k f kHS |∇ | (B ) For any f,g C∞(M), 4 ∈ 0 µ (∆f)(∆g) Hess , Hess = Kµ f, g . −h f giHS h∇ ∇ i (B ) There exists h : [0, ) M [0, ) with lim h(t, ) = 0 such that 5 ∞ × → ∞ t→0 · P f 2 eKt P f 2 h(t, ) Hess 2 + P Hess 2 , t 0, f C∞(M). t|∇ | − |∇ t | ≤ · k Ptf kHS tk f kHS ≥ ∈ 0
(C) Ricci Parallel manifolds. M is Ricci parallel if and only if
2 2 ∞ (∆f) Hessf Q f, f dµ =0, f C (M) Z −k kHS −h ∇ ∇ i ∈ 0 M holds for some constant symmetric linear map Q : T M T M, and in this case ic# = Q. They are also equivalent to each of the following statements:→ R
1 (C ) There exists a function h : [0, ) M [0, ) with lim t− 2 h(t, ) = 0 such that 1 ∞ × → ∞ t↓0 · Hess P Hess h(t, ) P Hess + Hess , t 0, f C2(M). k Ptf − t f k ≤ · tk f k k Ptf k ≥ ∈ b (C ) < , and for any x M, t 0, f C (M) and v , v T M, 2 kRk∞ ∞ ∈ ≥ ∈ 0 1 2 ∈ x t E E HessP f (v1, v2) Hessf (Wt(v1), Wt(v2)) = HessP − f Ws(v1), Ws(v2) ds. t − Z R t s 0 (C ) For any f C∞(M) and x M, 3 ∈ 0 ∈ Hess ∆Hess (v , v )=2 Hess (v , v ) 2 ic(v , Hess#(v )), v , v T M. ∆f − f 1 2 R f 1 2 − R 1 f 2 1 2 ∈ x (C ) For any x M and f C∞(M) with Hess (x) = 0, 4 ∈ ∈ 0 f (∆Hess )(v , v ) = Hess (v , v ), v , v T M. f 1 2 ∆f 1 2 1 2 ∈ x
Since a symmetric 2-tensor is determined by its diagonal, one may take v1 = v2 in (C2)-(C4).
To prove these results, we establish the following variational and derivative formulas for ic, see Remark 2.1 and Theorem 5.1 below. R
5 (D) Formulas of ic. Let ic and ic be the exact upper and exact lower bounds of ic. Then R R R R ic = inf µ (∆f)2 Hess 2 : f C∞(M),µ( f 2)=1 , R −k f kHS ∈ 0 |∇ | n o ic = sup µ (∆f)2 Hess 2 : f C∞(M),µ( f 2)=1 . R −k f kHS ∈ 0 |∇ | n o Moreover, let x M, v , v T M. For any f C4(M) with f(x)= v , Hess (x) = 0, ∈ 1 2 ∈ x ∈ b ∇ 1 f
(PtHessf HessPtf )(v1, v2) ( v2 ic)(v1, v1) = 2 lim − = ∆Hessf Hess∆f (v1, v2). ∇ R t↓0 t − The remainder of the paper is organized as follows. In Section 2, we present variational formulas of Bakry-Emery-Ricci upper and lower bounds, as well as integral characterizations of Einstein and Ricci parallel manifolds. In Section 3, we recall derivative and Hessian formulas of Pt developed from stochastic analysis. Using these formulas we estimate the Hessian of Pt in Section 4, and establish a formula for ic in Section 5. Finally, by applying results presented in Sections 3-5, we identify constant∇R curvature manifolds, Einstein manifolds, and Ricci parallel manifolds in Sections 6-8 respectively.
2 Characterizations of Bakry-Emery-Ricci curvature
2 V (x) 2 Let V C (M), µV (dx) = e µ(dx), and LV = ∆+ V . Then LV is symmetric in L (µV ). Consider∈ the Bakry-Emery-Ricci curvature ∇
ic := ic Hess . R V R − V
Definition 2.1. The manifold M is called V -Einstein if icV = K for some constant K R, while it is called ic parallel if ic = 0 (i.e. ic#R: T M T M is a constant map).∈ R V ∇R V R V → By the integral formula of Bochner-Weitzenb¨ock, we have
2 2 ∞ (2.1) (LV f) Hessf icV ( f, f) dµV =0, f C (M). Z −k kHS − R ∇ ∇ ∈ 0 M According to Theorem 2.1 below, this formula identifies the curvature ic , and provides R V sharp upper and lower bounds of icV , as well as integral characterizations of V -Einstein and ic parallel manifolds. R R V Theorem 2.1. Let K R be a constant, and let Q : T M T M be a symmetric continuous linear map. ∈ → (1) ic# = Q if and only if R V 2 2 ∞ (2.2) (LV f) Hessf Q f, f dµV =0, f C (M). Z −k kHS −h ∇ ∇ i ∈ 0 M Consequently, M is icV parallel if and only if (2.2) holds for some symmetric constant linear map Q : T M R T M. → 6 (2) For any V C2(M) and K R, ic K if and only if ∈ ∈ R V ≥
2 2 2 ∞ (2.3) (LV f) Hessf HS dµV K f dµV , f C0 (M); ZM n −k k o ≥ ZM |∇ | ∈ while ic K if and only if R V ≤
2 2 2 ∞ (2.4) (LV f) Hessf HS dµV K f dµV , f C0 (M). ZM n −k k o ≤ ZM |∇ | ∈
(3) M is V -Einstein if and only if
2 µV f µV (LV f)(LV g) Hessf , Hessg HS (2.5) |∇ | · −h i = µ f, g µ (L f)2 Hess 2 , f,g C∞(M). V h∇ ∇ i · V V −k f kHS ∈ 0 Moreover, for any constant K R, ic = K is equivalent to each of ∈ R V
2 2 2 ∞ (2.6) (LV f) Hessf K f dµV =0, f C (M), Z −k kHS − |∇ | ∈ 0 M and
∞ (2.7) (LV f)(LV g) Hessf , Hessg HS K f, g dµV =0, f,g C (M). Z −h i − h∇ ∇ i ∈ 0 M Remark 2.1. Theorem 2.1(2) provides the following variational formulas of the upper and lower bounds of ic . Let R V ic = sup ic (u,u): u TM, u =1 , ic = inf ic (u,u): u TM, u =1 . R V {R V ∈ | | } R V {R V ∈ | | } We have
ic = inf µ (L f)2 Hess 2 : f C∞(M),µ ( f 2)=1 , R V V V −k f kHS ∈ 0 V |∇ | n o ic = sup µ (L f)2 Hess 2 : f C∞(M),µ ( f 2)=1 . R V V V −k f kHS ∈ 0 V |∇ | n o To prove Theorem 2.1, we need the following lemma.
Lemma 2.2. For a continuous symmetric linear map Q : T M T M, if →
∞ (2.8) Q f, f dµV 0, f C0 (M), ZM h ∇ ∇ i ≥ ∈ then Q 0; that is, Qu,u 0 for u TM. ≥ h i ≥ ∈ V Proof. Using e Q replacing Q, we may and do assume that V = 0 so that µV = µ is the volume measure.
7 d (a) We first consider M = R for which we have Q =(qij)1≤i,j≤d with qij = qji for some d continuous functions qij on R . Thus,
d Qu, v = q u v , u,v Rd. h i ij i j ∈ i,jX=1
Without loss of generality, we only prove that Q(0) 0. Using the eigenbasis of Q(0), we ≥ may and do assume that Q(0) = diag q1, , qd . It suffices to prove ql 0 for 1 l d. For any f C∞(Rd), let { ··· } ≥ ≤ ≤ ∈ 0 f (x)= f φ (x), (φ (x)) := nx if i = l, (φ (x)) := n2x , n 1. n ◦ n n i i 6 n l l ≥ By (2.8),
d
0 qij(x)(∂ifn)(x)∂j fn(x)dx ≤ ZRd i,jX=1
4 2 3 = n qll(x)(∂lf) φn(x)+2n qjl(x) (∂lf)(∂jf) φn(x) ZRd ◦ { }◦ Xj6=l + n2 q (x) (∂ f)(∂ f) φ (x) dx ij { i j }◦ n i,jX6=l
3−d −1 2 2−d −1 = n qll φn (x)(∂lf) (x)+2n qjl φn (x) (∂lf)(∂jf) (x) ZRd ◦ ◦ { } Xj6=l + n1−d q φ−1(x) (∂ f)(∂ f) (x) dx, ij ◦ n { i j } i,jX6=l
−1 −1 where the last step is due to the integral transform x φn (x). Since φn (x) 0 as n , multiplying both sides by nd−3 and letting n 7→ we arrive at → →∞ →∞ 2 ∞ Rd ql(∂lf) (x)dx 0, f C0 ( ). ZRd ≥ ∈ Thus, q 0 as wanted. l ≥ (b) In general, for x0 M, we take a neighborhood O(x0) of x0 such that it is diffeo- d ∈ d d morphic to R with x0 corresponding to 0 R . Let ψ : O(x0) R with ψ(x0)=0bea diffeomorphism. Then under the local charts∈ induced by ψ, →
d Q (f ψ), (f ψ) dµ = q (x)(∂ f)(x)(∂ f)(x)dx, f C∞(Rd) h ∇ ◦ ∇ ◦ i ij i j ∈ 0 i,jX=1 holds for some symmetric matrix-valued continuous functional (qij)1≤i,j≤d. Therefore, by step (a), (2.8) implies d q (x)u u 0,x,u Rd. In particular, Q(x ) 0. i,j=1 ij i j ≥ ∈ 0 ≥ P
8 Proof of Theorem 2.1. (1) By the Bochner-Weitzenb¨ock formula, we have 1 (2.9) L f 2 f, L f = ic ( f, f)+ Hess 2 , f C∞(M). 2 V |∇ | − h∇ ∇ V i R V ∇ ∇ k f kHS ∈ 0 So, ic# = Q implies R V 1 L f 2 f, L f = Q f, f + Hess 2 . 2 V |∇ | − h∇ ∇ V i h ∇ ∇ i k f kHS
Integrating both sides with respect to µV proves (2.2). On the other hand, integrating both of (2.9) with respect to µV , we obtain (2.1). This together with (2.2) implies
# ∞ (2.10) icV ( f) Q f, f dµV =0, f C0 (M). ZRd R ∇ − ∇ ∇ ∈
# # Therefore, by Lemma 2.2 for icV Q replacing Q, we prove icV = Q. R − R # (2) By integrating both sides of (2.9) with respect to µV , we see that icV K implies # R ≥ (2.3). On the other hand, applying Lemma 2.2 to Q = icV K, if (2.3) holds then ic# K. Similarly, we can prove the equivalence of icR# −K and (2.4). Therefore, R V ≥ R V ≤ Theorem 2.1(2) holds. (3) By assertion (2), ic = K is equivalent (2.6). It remains to prove that (2.5) is R V equivalent to the Einstein property, since this together with the equivalence of icV = K and (2.6) implies the equivalence of (2.6) and (2.7). R R ∞ If M is V -Einstein, there exists a constant K such that icV = K. Let f,g C0 (M) 2 ∈ R ∈2 with µV ( f ) > 0, let fs = f + sg. Then there exists s0 > 0 such that µV ( fs ) > 0 for s [0,s ].|∇ By| (2.6), we have |∇ | ∈ 0 µ ((L f )2 Hess 2 ) h(s) := V V s −k fs kHS = K, s [0,s ]. µ ( f 2) ∈ 0 V |∇ s| So,
µ ( f 2)µ ((L f)(∆g) Hess , Hess ) µ ( f, g )µ ((L f)2 Hess 2 ) V |∇ | V V −h f gi − V h∇ ∇ i V V −k f kHS µ ( f 2)2 V |∇ | 1 = h′(0) = 0. 2 Therefore, (2.5) holds. On the other hand, for f C∞(M) with µ ( f 2) > 0, let ∈ 0 V |∇ | µ ((L f)2 Hess 2 ) K = V V −k f kHS R. µ ( f 2) ∈ V |∇ | By the equivalence of ic# = K and (2.6), it suffices to prove R V (2.11) µ (L g)2 Hess 2 = Kµ ( g 2), g C∞(M). V V −k gkHS V |∇ | ∈ 0 9 By the definition of K, this formula holds when g is a linear combination of f and 1. So, we assume that f,g and 1 are linear independent. In this case,
g := (1 s)f + sg, s [0, 1] s − ∈ satisfies µ ( g 2) > 0. Let V |∇ s| µ ((L g )2 Hess 2 ) K(s) := V V s −k gs kHS , s [0, 1]. µ ( g 2) ∈ V |∇ s| Then (2.5) implies 2 K′(s)= µ ( g 2)µ (L g )L (g f) Hess , Hess µ ( g 2)2 V |∇ s| V V s V − −h gs g−f iHS V |∇ s| n µ ( g , (g f) )µ (L g )2 Hess 2 =0, s [0, 1]. − V h∇ s ∇ − i V V s −k gs kHS ∈ o Therefore, µ ((L g)2 Hess 2 ) V V −k gkHS = K(1) = K(0) = K, µ ( g 2) V |∇ | that is, (2.11) holds as desired.
3 Derivative and Hessian formulas of Pt
In this section, by using the (horizontal) Brownian motion, we first formulate the heat semigroup Pt acting on tensors, then recall the derivative and Hessian formulas of Pt on functions.
3.1 Brownian motion and heat semigroup on tensors Consider the projection operator from the orthonormal frame bundle O(M) onto M:
π : O(M) M; πΦ= x if Φ O (M). → ∈ x Then for any a Rd and Φ =(Φi) O(M), ∈ 1≤i≤d ∈ d Φa := a Φi T M; i ∈ πΦ Xi=1
and for any v TπΦM, ∈ d Φ−1v := v, Φi e Rd, h i i ∈ Xi=1 where e is the canonical orthonormal basis of Rd. { i}1≤i≤d
10 For any a Rd and Φ O(M), let Φ(s) be the parallel transport of Φ along the geodesic s exp[sΦa],s∈ 0. We have∈ 7→ ≥ d H (Φ) := Φ(s) T O(M), a ds s=0 ∈ Φ
which is a horizontal vector field on O(M). Let Hi = Hei . Then (Hi)1≤i≤d forms the canonical orthonormal basis for the space of horizontal vector fields:
d H = a H , a =(a ) Rd. a i i i 1≤i≤d ∈ Xi=1 We call
d 2 (3.1) ∆O(M) := Hi Xi=1 the horizontal Laplacian on O(M). For any smooth n-tensor , T O(M) i1 in n d i (Φ) := (Φ , , Φ ) R , Φ=(Φ )1≤i≤d O(M) T T ··· 1≤i1,··· ,in≤d ∈ ⊗ ∈ gives rise to a smooth nRd-valued function on O(M). Let ∆ be the Bochner Laplacian ∆. For any x M, v , ⊗, v T M and Φ O (M), we have ∈ 1 ··· n ∈ x ∈ x (∆ )(v , , v )= ∆ O(M)(Φ) (Φ−1v , , Φ−1v ), T 1 ··· n { O(M)T } 1 · n (3.2) O(M) −1 −1 ( v )(v1, , vn)=( H −1 )(Φ v1, , Φ vn), v TxM. ∇ T ··· ∇ Φ v T ··· ∈ Now, consider the following SDE on O(M):
d (3.3) dΦ = H(Φ ) dB := H (Φ ) dBi, t t ◦ t i t ◦ t Xi=1
i where Bt := (Bt)1≤i≤d is the d-dimensional Brownian motion on a complete probability space F P F B (Ω, , ) with natural filtration t := σ(Bs : s t) (by convention, we always take the completion of a σ field). The solution is called the≤ Horizonal Brownian motion on O(M). Let ζ be the life time of the solution. When ζ = (i.e. the solution is non-explosive) we call the manifold M stochastically complete. It is∞ the case when
(3.4) ic c(1 + ρ2) R ≥ − for some constant c> 0, where ρ is the Riemannian distance to some fixed point, see the proof of Proposition 3.1 blow. See also [8] and references within for the stochastic completeness under weaker conditions. Let X := πΦ for t [0,ζ). Then (X ) solves the SDE t t ∈ t t∈[0,ζ) (3.5) dX = Φ (X ) dB , t t t ◦ t 11 and is called the Brownian motion on M. For any x M, let Xt(x) be the solution of (3.5) with X = x. Note that X (x) does not depend on the∈ choice of the initial value Φ O (M) 0 t 0 ∈ x for (3.3), and for fixed initial value Φ0 Ox(M), both Φt(Xt) and Bt are measurable with respect to F X := σ(X : s t). Therefore,∈ F B = F X , t 0. When the initial point x is t s ≤ t t ≥ clearly given in the context, we will simply denote Xt(x) by Xt. Let := Φ Φ−1 : T M T M kt t 0 X0 → Xt be the stochastic parallel transport along Xt, which also does not depend on the choice of Φ . For any smooth n-tensor with compact support, let 0 T (3.6) (P )(v , , v )= E 1 ( v , , v ) , v , , v T M. tT 1 ··· n {t<ζ}T kt 1 ··· kt n 1 ··· n ∈ x As is well known in the function (i.e. 0-tensor) setting, by (3.1), the first formula in (3.2), (3.3) and Itˆo’s formula we have the forward/backward Kolmogorov equations 1 1 (3.7) ∂ P = ∆P = P ∆ , C∞. t tT 2 tT 2 t T T ∈ 0 For later use, we present the following exponential estimate of the Brownian motion under condition (3.4). Proposition 3.1. Assume (3.4). Then there exist constants r, c(r) > 0 such that
−rt −rt 2 2 c(r)(1 e ) (3.8) E exp[e ρ (Xt)] exp ρ (x)+ − , t 0, x M. ≤ h r i ≥ ∈ Consequently, if log ic (3.9) lim k∇R k =0, ρ→∞ ρ2
then for any ε (0, 1) and p 1, there exists a positive function C C([0, )) such that ∈ ≥ ε,p ∈ ∞ 2 (3.10) E ic p(X (x)) eερ (x)+Cε,p(t), t 0, x M. k∇R k t ≤ ≥ ∈ Proof. Let ρ be the Riemannian distance to a fixed point o M. By the Laplacian compar- ison theorem, (3.4) implies ∈ ∆ρ c (ρ + ρ−1) ≤ 1 outside o cut(o) for some constant c > 0, where cut(o) is the cut-locus of o. By Itˆo’s { } ∪ 1 formula of ρ(Xt) given in [10], this gives
dρ(X )2 c 1+ ρ(X )2 dt +2ρ(X )db t ≤ 2{ t } t t for some constant c2 > 0 and an one-dimensional Brownian motion bt. By Itˆo’s formula, for any r > 2+ c2 there exists a constant c(r) > 0 such that
d exp[e−rtρ2(X )] exp[e−rtρ2(X )] (c + c ρ2(X ))e−rt re−rtρ2(X ) t ≤ t 2 2 t − t 12 −2rt 2 −rt + 2e ρ (Xt) dt + 2e ρ(Xt)dbt exp[e−rtρ2(X )] c(r)e−rtdt + 2e−rtρ(X )db . ≤ t t t Therefore, (3.8) holds. On the other hand, by (3.9) we may find a positive function c C([0, )) such that p/ε ∈ ∞ ic# p/ε exp e−tρ2 + c (t) , t 0. k∇R k ≤ p/ε ≥ Combining this with (3.8) for r = 1, we obtain
p p/ε ε E ic (Xt(x)) E ic (Xt) k∇R2 k ≤ε k∇R2 k eρ (x)+c(1)+cε/p(t) eερ (x)+εc(1)+εcp/ε (t). ≤ ≤ Therefore, (3.10) holds for Cε,p(t) := ε[c(1) + cp/ε(t)].
3.2 Derivative formula of Pt In this subsection, we assume h(r) (3.11) ic h(ρ) for some positive h C([0, )) with lim =0. R ≥ − ∈ ∞ r→∞ r2 Let W : T M T M be defined in (1.2). By (3.11) and Proposition 3.1, we have t x → Xt p (3.12) E sup Ws < , t> 0. s∈[0,t] k k ∞
We have (see e.g. [7, 8])
(3.13) P f(x)= E f(X (x)), W (v) , t 0, f C1(M). ∇v t h∇ t t i ≥ ∈ b Following the idea of [7, 14], it is standard to establish the Bismut type formula using 1 (3.13). By ∂tPtf = 2 ∆Ptf, (3.5) and Itˆo’s formula, we have dP f(X )= P f(X ), Φ dB , t−s s h∇ t−s s s si so that
t (3.14) f(Xt(x)) = Ptf(x)+ Pt−sf(Xs(x)), Φs(x)dBs . Z0 h∇ i
In particular, Pt−sf(Xs) is a martingale. Next, by (3.13) and the Markov property, we have
P f(X ), W (v) = E f(X ), W (v) F B , s [0, t], ∇ t−s s s ∇ t t s ∈ which is again a martingale. Indeed, according to e.g. [19, (2.2.7)], we have
(3.15) d P f(X ), W (v) = Hess − (Φ dB , W (v)), s [0, t]. h∇ t−s s s i Pt sf s s s ∈ 13 1 So, for any adapted process h C ([0, t]) such that h0 =0, ht = 1, (3.13), (3.14) and (3.15) imply ∈
t t ˙ ˙ E f(Xt(x)) hs Ws(v), Φs(x)dBs = E hs Ws(v), Pt−sf(Xs(x)) ds Z0 h i Z0 h ∇ i t t = E hs Ws(v), Pt−sf(Xs(x)) hsd Ws(v), Pt−sf(Xs(x)) h ∇ i 0 − Z h ∇ i 0 = E W (v), f(X (x)) = P f(x). h t ∇ t i ∇v t Therefore,
t E ˙ 1 (3.16) vPtf(x)= f(Xt(x)) hs Ws(v), Φs(x)dBs , t> 0, x M, f Cb (M). ∇ Z0 h i ∈ ∈ This type formula is named after J.-M. Bismut, K.D. Elworthy and X.-M. Li because of their pioneering work [5] and [7]. The present version is due to [14] and has been applied in [2, 15] to derive gradient estimates using local curvature conditions.
3.3 Hessian formula of Pt
To calculate HessPtf , we introduce the following doubled damped parallel transport Wt(v1, v2) for v , v T M: 1 2 ∈ x t −1 (2) 1 −1 ˜ # # (2) Φt(x) Wt (v1, v2)= Φs ( ic )(Ws(v2))Ws(v1) ic (Ws (v1, v2)) ds 2 Z0 ∇R − R (3.17) t −1 + Φs(x) (Φs(x)dBs, Ws(v2))Ws(v1), t 0, Z0 R ≥ where the cycle derivative ˜ ic# is defined by ∇R (3.18) ( ˜ ic#)(u )u ,u := ( ic)(u ,u ) ( ic)(u ,u ) ( ic)(u ,u ) h ∇R 2 1 3i ∇v3 R 1 2 − ∇u1 R 2 3 − ∇u2 R 1 3 for u1,u2,u3 TyM,y M. According to Proposition 3.1 and (3.12), conditions (3.9) and (3.11) imply ∈ ∈ E (2) p sup Ws < , p 1,t> 0. s∈[0,t] k k ∞ ≥
Proposition 3.2 ([1, 12]). Assume (3.9) and (3.11). Then for any f C2(M) and v , v ∈ b 1 2 ∈ TxM,
(3.19) Hess (v , v )= E Hess (W (v ), W (v )) + f(X (x)), W (2)(v , v ) . Ptf 1 2 f t 1 t 2 h∇ t t 1 2 i Proof. Let v (s) be the parallel transport of v along the geodesic s exp[sv ],s 0. 2 2 7→ 1 ≥ According to (3.10), we define the following covariant derivative of Wt: d W (2)(v , v ) := W (v )= W (v (s)) . t 1 2 ∇v1 t 2 ds t s s=0
14 By (3.13),
Hess (v , v )= E Hess (W (v ), W (v )) + f(X (x)), W (2)(v , v ) . Ptf 1 2 f t 1 t 2 h∇ t t 1 2 i (2) F X F B It remains to prove that Wt satisfies (3.17). Since in the present setting t = t , this follows from formula (7) in [1], see also (3.1) in [12]. In the same spirit of deducing the Bismut type derivative formula (3.16) from (3.13), Bismut type Hessian formulas of Ptf have been presented in [1, 7, 12] by using (3.19). In Section 4 we will use the following local version of Hessian formula, which follows from [1, Theorem 2.1] and [1, Proof of Theorem 3.1] for e.g. D1 = B(x, 1),D2 = B(x, 2), where B(x, r) is the open geodesic ball at x with radius r.
Proposition 3.3 ([1]). Let M be a complete noncompact Riemannian manifold. Let
τ (x) = inf t 0 : X (x) ∂B(x, i) , i =1, 2. i { ≥ t ∈ }
There exists a positive function C C(M) such that for any x M, v1, v2 TxM with v , v 1, and f B (M), ∈ ∈ ∈ | 1 | 2| ≤ ∈ b E (3.20) HessPtf (v1, v2)= Pt−t∧τ1(x)f(Xt∧τ1(x)(x))Mt + Pt−t∧τ2(x)f(Xt∧τ2(x)(x))Nt , t> 0. holds for some adapted continuous processes (Mt, Nt)t≥0 determined by (Xt(x))0≤t≤τ2(x) such that C(x) (3.21) E N + M , t> 0. | t| | t| ≤ t 1 ∧ 4 Hessian estimates and applications
In this section, we first present Hessian estimates of Pt for Einstein and Ricci parallel mani- folds, then apply these results to describe the lower and upper bounds of the Ricci curvature. Recall that for any x M and f C2(M), ∈ ∈ Hess (x) := sup Hess (u, v) : u, v T M, u , v 1 , k f k {| f | ∈ x | | | | ≤ } d Hess 2 (x) := Hess (Φi, Φj)2, Φ=(Φi) O (M). k f kHS f 1≤i≤d ∈ x i,jX=1
Theorem 4.1. Let M be a Ricci parallel manifold with ∞ < . Then for any x M, t 0, f C (M) and v , v T M, kRk ∞ ∈ ≥ ∈ 0 1 2 ∈ x Hess (v , v ) E Hess (W (v ), W (v )) Ptf 1 2 − f t 1 t 2 (4.1) t E = HessP − f Ws(v1), Ws(v2) ds, Z R t s 0
where Hess − is defined in (1.1) for T = Hess − . Consequently: R Pt sf Pt sf 15 (1) If ic K, then for any f C2(M), R ≥ ∈ b (4.2) Hess e(kRk∞−K)tP Hess , t 0. k Ptf k ≤ tk f k ≥ (2) If ic = K, then R (4.3) Hess 2 e2(kRk∞−K)tP Hess 2 , t 0. k Ptf kHS ≤ tk f kHS ≥
2 Proof. We fix t> 0 and f Cb (M). Let d be the exterior differential. By e.g. [19, (2.2.6)] we have ∈ 1 (4.4) d(dP f)(X )= (dP f)(X )+ ic( , P f(X ))ds, s [0, t]. t−s s ∇ΦsdBs t−s s 2R · ∇ t−s s ∈ Equivalently,
1 t t −1 f X −1 P f −1 ic# P f X s # B . (4.5) Φt ( t) = Φ0 t + Φs ( t−s ( s))d + HessPt−sf (Φsd s) ∇ ∇ 2 Z0 R ∇ Z0
On the other hand, since ˜ ic# = 0, (3.17) becomes ∇R t t −1 (2) −1 1 −1 # (2) Φt Wt (v1, v2)= Φs (ΦsdBs, Ws(v2))Ws(v1) Φs ic (Ws (v1, v2))ds. Z0 R − 2 Z0 R Combining this with (??), we obtain
t E (2) E (4.6) f(Xt), Wt (v1, v2) = tr HessPt−sf ( , ( , Ws(v2))Ws(vs)) . h∇ i Z0 n · R · o Plugging (4.6) into (3.19) gives
Hess (v , v ) E Hess (W (v ), W (v )) Ptf 1 2 − f t 1 t 2 t E , W v W v , # s = tr ( s( 2)) s( 1) HessPt−sf ( ) d Z0 R · · t E = HessP − f (Ws(v1), Ws(v2))ds. Z R t s 0 Therefore, (4.1) holds. Below we prove (4.2) and (4.3) for Ricci parallel and Einstein manifolds respectively. (a) (4.1) and ic K imply (4.2). If ic K, then (1.2) implies R ≥ R ≥ 1 W (v) e− 2 Kt v . | t | ≤ | | So, according to (4.1), for any s> 0 we have
s −Ks −Kr HessPsf e Ps Hessf + ∞ e Pr HessPs−rf dr. k k ≤ k k kRk Z0 k k
16 Letting φ(s) = e−K(t−s)P Hess , s [0, t], t−sk Psf k ∈ we obtain s −K(t−s) −sK −rK φ(s) e Pt−s e Ps Hessf + ∞ e Pr HessPs−rf dr ≤ k k kRk Z0 k k t −Kt −K(t+r−s) e Pt Hessf + ∞ e Pt+r−s HessPs−rf dr. ≤ k k kRk Z0 k k Using the change of variable θ = s r, we arrive at − s φ(s) φ(0) + ∞ φ(θ)dθ, s [0, t]. ≤ kRk Z0 ∈ By Gronwall’s lemma, this implies
φ(t) φ(0)ekRk∞t, ≤ which is equivalent to (4.2). K − 2 s (b) Let ic = K. Then (1.2) implies Ws(v) = e v. So, for x M, v TxM and Φ O (M),R (4.1) implies ∈ ∈ 0 ∈ x Hess (v, Φk) = e−KtE Hess ( v, Φk) Ptf 0 f kt t (4.7) t E −sK k + e HessP − f (Φ , v)ds. Z R t s s ks 0 Let −K(t−s) # k φk(s) = e E Hess (Φ ) , s [0, t], 1 k d. k Psf t−s k ∈ ≤ ≤ By (4.7) and the Markov property, for any 0 s
Ik,v(s1,s2) E −(t−s1)K k −K(t−s2)E k F x := e HessPs1 f ( t−s1 v, Φt−s1 ) e HessPs2 f ( t−s2 v, Φt−s2 ) t−s1 k − k s1−s2
v e−(t−s1)K E e−rK Hess# (Φk ) dr ∞ Ps1−rf t−s1+r ≤| | kRk Z0 | | s1 = v e−K(t−θ)E Hess# (Φk ) dθ ∞ Pθf t−θ | |kRk Zs2 | | s2 = v ∞ φk(θ)dθ, | |kRk Zs1
17 where we have used the change of variable θ = s r. Then 1 −
φk(s1) φk(s2) sup Ik,v(s1,s2) − ≤ |v|≤1 s2 ∞ φk(θ)dθ, 0 s2 s1 t. ≤ kRk Zs1 ≤ ≤ ≤ By Gronwall’s lemma, this implies
Hess# (Φk) = φ (t) ekRk∞tφ (0) = e(kRk∞−K)tE Hess#(Φk) , 1 k d. | Ptf 0 | k ≤ k | f t | ≤ ≤ Therefore,
d d 2 Hess 2 = Hess# (Φk) 2 e2(kRk∞−K)t E Hess#(Φk) k Ptf kHS | Ptf 0 | ≤ | f t | Xk=1 Xk=1 e2(kRk∞−K)tP Hess 2 . ≤ tk f kHS
Next, we apply the above results to characterize the lower and upper bounds of ic for R Ricci parallel manifolds.
Theorem 4.2. Let M be a Ricci parallel manifold. Then for any constant K R, the following statements are equivalent each other: ∈
(1) ic K. R ≥ (2) For any f C∞(M) and t 0, ∈ 0 ≥ eKt e2(K−kRk∞)t − Hess 2 P f 2 eKt P f 2. 2 K k Ptf k ≤ t|∇ | − |∇ t | kRk∞ − (3) For any f C∞(M) and t 0, ∈ 0 ≥ t Ks 2(K−kRk∞)s Kt 2 e e 2 2 e 1 2 HessP f − ds Ptf (Ptf) − Ptf . k t k Z 2 K ≤ − − K |∇ | 0 kRk∞ − (4) For any f C∞(M) and t 0, ∈ 0 ≥ 1 e−Kt P f 2 (P f)2 − P f 2 t − t − K t|∇ | t 2(kRk∞−K)s −Ks 2 2(K−kRk∞)t e e HessPtf e − ds. ≤ −k k Z 2 ∞ K 0 kRk −
18 Proof. (a) (1) (2). Let t> 0 and f C2(M). By (3.5) and Itˆo’s formula, we have ⇒ ∈ b
2 1 2 d Pt−sf (Xs)= ∆ Pt−sf (Xs) Pt−sf, ∆Pt−sf (Xs) ds (4.8) |∇ | 2 |∇ | − h∇ ∇ i +2 P f 2(X ), Φ dB , s [0, t]. h∇|∇ t−s | s s si ∈ By the Bochner-Weitzenb¨ock formula and ic K, we obtain R ≥ 1 ∆ P f 2(X ) P f, ∆P f (X ) 2 |∇ t−s | s − h∇ t−s ∇ t−s i s 2 = ic( Pt−sf, Pt−sf)(Xs)+ HessPt−sf HS(Xs) R ∇ 2 ∇ k 2 k K P f (X )+ Hess − (X ). ≥ |∇ t−s | s k Pt sf kHS s Then (4.8) implies
2 2 2 2 d P f (X ) K P f + Hess − (X )ds +2 P f (X ), Φ dB |∇ t−s | s ≥ |∇ t−s | k Pt sf kHS s h∇|∇ t−s | s s si for s [0, t]. Combining this with (4.2), we arrive at ∈ t 2 Kt 2 K(t−s) 2 Pt f e Ptf e Ps HessPt−sf HSds |∇ | − |∇ | ≥ Z0 k k t K(t−s) −2(kRk∞−K)s 2 e e HessPtf ds ≥ Z0 k k eKt e2(K−kRk∞)t = − Hess 2. 2 K k Ptf k kRk∞ − (b) (2) implies (3) and (4). By (3.5) and Itˆo’s formula, we have
d(P f)2(X )= P f 2(X )ds + P f 2(X ), Φ dB , s [0, t]. t−s s |∇ t−s | s h∇| t−s | s s si ∈ So,
t 2 2 2 (4.9) Ptf (Ptf) = Ps Pt−sf ds. − Z0 |∇ | Combining this with (2) and (4.2), we obtain
P f 2 (P f)2 t − t t Ks 2(K−kRk∞)s 2 Ks e e 2 Ptf e + − HessP f ds. ≥ Z |∇ | 2 K k t k 0 n kRk∞ − o Then (3) is proved. Similarly, (4.2) and (2) imply
e−KsP f 2 P P f 2 t|∇ | − t−s|∇ s | 1 e(K−2kRk∞)s − P Hess 2 ≥ 2 K t−sk Psf k kRk∞ − 19 e2(kRk∞−K)s e−Ks e2(K−kRk∞)t Hess 2 − , ≥ k Ptf k · 2 K kRk∞ − which together with (4.9) gives (4). ∞ (c) Each of (3) and (4) implies (1). For v TxM with v = 1, take f C0 (M) such that ∈ | | ∈ f(x)= v, Hess (x)=0. ∇ f We have
t Ks 2(K−kRk∞)s 2 e e (4.10) lim HessPtf − ds =0. t↓0 k k Z 2 K 0 kRk∞ − On the other hand, by the Bochner-Weitzenb¨ock formula we have (see [19, Theorem 2.2.4]), 1 P f p(x) P f p(x) (4.11) ic(v, v) = lim t|∇ | − |∇ t | , p> 0. 2R t↓0 pt Combining this with (3), (4.9) and (4.10), we obtain
Kt P f 2 (P f)2 e −1 P f 2 0 2 lim t − t − K |∇ t | ≤ t↓0 t2 t 2 2 Ks 2 = lim Ps Pt−sf e Ptf (x)ds = ic(v, v) K, t↓0 t2 Z |∇ | − |∇ | R − 0 Therefore, (3) implies (1). Similarly, (4) also implies (1). The following result provides corresponding characterizations for the Ricci upper bound. Theorem 4.3. Let M be a Ricci parallel manifold. Then for any constant K R, the following are equivalent each other: ∈ (1) ic K. R ≤ (2) For any f C∞(M) and t 0, ∈ 0 ≥ e(2kRk∞−K)t 1 − dP Hess 2 P f 2 eKt P f 2. 2 K tk f k ≥ t|∇ | − |∇ t | kRk∞ − (3) For any f C∞(M) and t 0, ∈ 0 ≥ t (2kRk∞−K)t−Ks 2(kRk∞−K)s 2 e e dPt Hessf − ds k k Z 2 K 0 kRk∞ − eKt 1 P f 2 (P f)2 − P f 2. ≥ t − t − K |∇ t |
(4) For any f C∞(M) and t 0, ∈ 0 ≥ −Kt t 2(kRk∞−K)s −Ks 2 2 1 e 2 2 e e Ptf (Ptf) − Pt f dPt Hessf − ds. − − K |∇ | ≥ − k k Z 2 K 0 kRk∞ − 20 Proof. By using
(4.12) Hess 2 d Hess 2, k f kHS ≤ k f k the proof is completely similar to that of Theorem 4.2. For instance, below we only show the proof of (1) implying (2). By ic K and (4.8), we have R ≤ 2 2 2 2 d P f (X ) K P f + Hess − (X )ds +2 P f (X ), Φ dB |∇ t−s | s ≤ |∇ t−s | k Pt sf kHS s h∇|∇ t−s | s s si for s [0, t]. Combining this with (4.2) and (4.12), we arrive at ∈ t 2 Kt 2 K(t−s) 2 Pt f e Ptf d e Ps HessPt−sf ds |∇ | − |∇ | ≤ Z0 k k t K(t−s) 2(kRk∞−K)(t−s) 2 d e e Pt Hessf ds ≤ Z0 k k e(2kRk∞−K)t 1 = − P Hess 2. 2 K tk f k kRk∞ − Then (1) implies (2). We therefore omit other proofs.
5 Formula of ic ∇R Theorem 5.1. For any x M, v , v T M and f C4(M) with f(x)= v , Hess (x)= ∈ 1 2 ∈ x ∈ b ∇ 1 f 0, there holds
(PtHessf HessPtf )(v1, v2) (5.1) ( v2 ic)(v1, v1) = 2 lim − = ∆Hessf Hess∆f (v1, v2). ∇ R t↓0 t −
Consequently, M is Ricci parallel if and only if ∆Hessf = Hess∆f holds at any point x M and f C∞(M) with Hess (x)=0. ∈ ∈ 0 f ∞ When f C0 (M), the second equation in (5.1) follows from (3.7). By a standard ∈ 2 approximation argument, this equation holds for all f Cb (M). So, it suffices to prove the first equation or the formula (5.2) below for x M and∈ f C∞(M) with Hess (x) = 0. ∈ ∈ 0 f Here, we prove both of them by using analytic and probabilitstic arguments respectively, since each proof has its own interest. Analytic Proof. For any x M and f C∞(M) with Hess (x)=0, we intend to prove ∈ ∈ 0 f (5.2) ( ic)( f, f) = (∆Hess )( f, v) Hess ( f, v), v T M. ∇vR ∇ ∇ f ∇ − ∆f ∇ ∈ x According to the Bochner-Weitzenb¨ock formula, we have
(5.3) ic#( f) = ∆ f ∆f, R ∇ ∇ −∇
21 where ∆ f := Φ(∆ O(M))(Φ) is independent of Φ O(M). Consequently, ∇ O(M)∇f ∈ 1 (5.4) ic( f, f)= ∆ f 2 ∆f, f Hess 2 . R ∇ ∇ 2 |∇ | − h∇ ∇ i−k f kHS
Since Hessf (x) = 0, (5.4) and (5.3) imply that at point x, ( ic)( f, f)= ic( f, f) ∇R ∇ ∇ ∇{R ∇ ∇ } 1 = ∆ f 2 Hess# ( f) Hess#( ∆f) 2 Hess Hess 2∇ |∇ | − ∆f ∇ − f ∇ − k f kHS∇k f k 1 1 = ∆ f 2 ic#( f 2) Hess# ( f) 2 {∇|∇ | } − 2R ∇|∇ | − ∆f ∇ = ∆ Hess#( f) Hess# ( f) { f ∇ } − ∆f ∇ = (∆Hess )#( f) + Hess#(∆ f)+2tr ( Hess#)(Hess#( )) Hess# ( f) f ∇ f ∇ ∇· f f · − ∆f ∇ = (∆Hess )#( f) Hess# ( f). f ∇ − ∆f ∇ Therefore, (5.2) holds. Probabilistic Proof. We first consider bounded ic and , then extend to the general case by using Proposition 3.3. ∇R R
(a) Assume that ∞ + ic ∞ < . Let x M and v1, v2 TxM. We take f C4(M) such that kRkf(x) = vk∇Randk Hess (∞x) = 0. Below∈ we only consider∈ functions taking∈ b ∇ 1 f value at point x. Since Hessf (x) = 0, there exists a constant c> 0 such that
t 2 1 2 (5.5) Pt Hessf HS(x)= Ps∆ Hessf HS(x)ds ct, t 0. k k 2 Z0 k k ≤ ≥
Then there exists a constant c1 > 0 such that
2 (5.6) Ps Hessf Ps Hessf c1√s, s [0, 1]. k k ≤ q k k ≤ ∈ Since f(x)= v and P f(x) is smooth in s, this together with (4.5) yields ∇ 1 ∇ s (5.7) E f(X ) v c s, s [0, 1] |∇ s −ks 1| ≤ 2 ∈ for some constant c2 > 0. Moreover, by (1.2) there exists a constant c3 > 0 such that (5.8) W (v ) v c s, s [0, 1], i =1, 2. | s i −ks i| ≤ 3 ∈ Combining (5.6)-(5.8) with (3.10), (3.17) and (3.18), for small t> 0 we arrive at
t (2) 1 # E f(Xt), W (v1, v2) = E ( ˜ ic )(Ws(v2))Ws(v1), f(Xs) ds h∇ t i 2 Z ∇R ∇ i 0 d t (5.9) E i i + HessP − f Φ , (Φ , Ws(v2))Ws(v1) ds Z t s s R s Xi=1 0 1 1 =o(t)+ ( ˜ ic#)(v )v , v t = o(t) ( ic)(v , v )t, 2h ∇R 2 1 1i − 2 ∇v2 R 1 1 22 where the last step follows from (3.18). On the other hand, by (3.19), (5.6) and (5.8), there exist a constant c4 > 0 such that for small t> 0, E f(X ), W (2)(v , v ) = Hess (v , v ) EHess (W (v ), W (v )) h∇ t t 1 2 i Ptf 1 2 − f t 1 t 2 = Hess (v , v ) P Hess (v , v )+O(t)P Hess Ptf 1 2 − t f 1 2 tk f k = Hess (v , v ) P Hess (v , v )+o(t). Ptf 1 2 − t f 1 2 Combining this with (5.9) we derive the desired the first equation in (5.1).
(b) In general, let M be a complete non-compact Riemannian manifold. For fixed x M we take D = B(x, 4). Let f C∞(D¯) with f = 0, f = 1, f = 1 on ∂D∈and ∈ |∂D |B(x,3) |∇ | f > 0 in D. Then (D, f −2g) is a complete Riemannian manifold. We use superscript D to denote quantities on this manifold, for instance, D is the Riemannian tensor on (D, f −2g). Then both D and D icD are bounded. So, byR (a), for P D the heat semigroup on D, R ∇ R t D D D (P Hess Hess D )(v1, v2) t f Pt f (5.10) ( v2 ic)(v1, v1) = lim − . ∇ R t↓0 t D Since f = 1 in B(x, 3), we may construct the horizontal Brownian motion Φt (y) on D with ΦD(y) = Φ O (M) such that 0 0 ∈ y ΦD(y) = Φ (y), t τ (y), XD(y)= X (y), t τ (y), t t ≤ 3 t t ≤ 3 where τ (y) = inf t 0 : X (y) ∂B(x, 3) . 3 { ≥ t ∈ } Noting that P f(y)= E[f(X (y))1 ], P Df(y)= E[f(XD(y))], f B (M), t t {t<ζ} t t ∈ b where ζ is the life time of Xt, we obtain P f(y) P D(y) f P(τ (y) t), t 0, f B (M). | t − t |≤k k∞ 3 ≤ ≥ ∈ b Combining this with [3, Lemma 2.3], we may find constants c1,c2 > 0 such that (5.11) P f(y) P Df(y) c f e−c2/t, t (0, 1],y B(x, 2). | t − t | ≤ 1k k∞ ∈ ∈ Consequently, (5.12) P Hess (v , v ) P DHess (v , v ) c Hess e−c2/t, t (0, 1]. | t f 1 2 − t f 1 2 | ≤ 1k f k∞ ∈ D Moreover, since Xt = Xt B(x, 2) before time τ2, Proposition 3.3 and (5.11) imply that at point x, ∈ D HessPtf (v1, v2) Hess D (v1, v2) | − Pt f | E ( P f(X ) P D f(XD ) M ≤ | t−t∧τ1 t∧τ1 − t−t∧τ1 t∧τ1 |·| t| + E P f(X ) P D f(XD ) ) N | t−t∧τ2 t∧τ2 − t−t∧τ2 t∧τ2 | | t| C(x) c e−c2/t f , t (0, 1]. ≤ 1 k k∞ t ∈ Combining this with (5.10) and (5.12), we prove the first equation in (5.1).
23 6 Identification of constant curvature
Theorem 6.1. Let k R. Then each of the following assertions is equivalent to ec = k: ∈ S (1) For any t 0 and f C∞(M), ≥ ∈ 0 1 (6.1) Hess = e−dktP Hess + (1 e−dkt)(P ∆f)g. Ptf t f d − t
(2) For any f C∞(M), ∈ 0 (6.2) Hess ∆Hess =2k(∆f)g 2dkHess . ∆f − f − f
∞ (3) For any x M,u TxM and f C0 (M) with Hessf (x)= u u (i.e. Hessf (v1, v2)= u, v u, v∈ , v , v ∈ T M), ∈ ⊗ h 1ih 2i 1 2 ∈ x (6.3) Hess ∆Hess (v, v)=2k u 2 v 2 u, v 2 , v T M. ∆f − f | | | | −h i ∈ x (4) For any f C∞(M), ∈ 0 1 (6.4) ∆ Hess 2 Hess , Hess Hess 2 =2k d Hess 2 (∆f)2 . 2 k f kHS −h ∆f f iHS −k∇ f kHS k f kHS − To prove this result, we need the following lemma where T is defined in (1.1). R Lemma 6.2. If the sectional curvature ec = k for some constant k, then for any symmetric 2-tensor T , S T = ktr(T )g kT. R − Proof. Let T˜ = ktr(T )g kT. Since both T and T˜ are symmetric, it suffices to prove − R (6.5) ( T )(v, v)= T˜(v, v), v T , v =1, x M. R ∈ x | | ∈ Let v T M with v = 1. By the symmetry of T , there exists Φ = (Φi) O (M) such ∈ x | | 1≤i≤d ∈ x that T #(Φi)= λ Φi, 1 i d i ≤ ≤ holds for some constants λ , 1 i d. Then ec = k implies i ≤ ≤ S d d ( T )(v, v)= (Φi, v)v, T #(Φi) = λ (Φi, v)v, Φi R R i R Xi=1 Xi=1 d = k λ 1 Φi, v 2 = ktr(T ) kT (v, v)= T˜(v, v). i −h i − Xi=1 Therefore, (6.5) holds.
24 Proof of Theorem 6.1. Obviously, (3) follows from (2). Next, by (3.7) and taking derivative of (6.1) with respect to t at t = 0, we obtain (6.2). So, (1) implies (2). Moreover, by chain rule we have 1 (6.6) ∆ Hess 2 = ∆Hess , Hess + Hess 2 . 2 k f kHS h f f iHS k∇ f kHS Then (4) follows from (2) and the identity (δf)g, Hess = (∆f)2. To complete the proof, h f iHS below we prove “ ec = k (1)”, “(3) ec = k ” and “(4) ec = k ” respectively. (a) ec = k S (1).⇒ Let ec = k⇒. ThenS ic = (d 1)⇒k. S By (1.2), (4.1), we have S K ⇒ S R − W (v) = e− 2 s v, s 0, v T M, and for any x M, v , v T M, s ks ≥ ∈ ∈ 1 2 ∈ x t −Kt −Ks HessPtf (v1, v2) = e PtHessf (v1, v2)+ e Ps( HessPt−sf )(v1, v2)ds. Z0 R
Noting that ∆Pt−sf = Pt−s∆f and s v1, s v2 = v1, v2 , this together with Lemma 6.2 gives hk k i h i
−Kt HessPtf (v1, v2) e PtHessf (v1, v2) t − −Ks = e Ps k v1, v2 Pt−s∆f kPs(HessP − f )(v1, v2) ds, t 0. Z h i − t s ≥ 0 Therefore,
t −(K+k)t −(K+k)s HessPtf (v1, v2) = e PtHessf (v1, v2)+ k v1, v2 e Pt∆fds h i Z0 1 e−dkt = e−dktP Hess (v , v )+ − (P ∆f) v , v . t f 1 2 d t h 1 2i So, (6.1) holds. (b) (3) ec = k. By taking u = 0, (3) implies that for any f C∞(M) with ⇒ S ∈ 0 Hessf (x) = 0, (∆Hess Hess )(v, v)=0, v T M. f − ∆f ∈ x By the symmetry of ∆Hess Hess , this is equivalent to f − ∆f (∆Hess Hess )(v , v )=0, v , v T M. f − ∆f 1 2 1 2 ∈ x ∞ So, for any v1, v2 TxM, by taking f C0 (M) such that f(x) = v1 and Hessf (x) = 0, we deduce from Theorem∈ 5.1 that ∈ ∇
( ic)(v , v )= ∆Hess Hess (v , v )=0. ∇v2 R 1 1 f − ∆f 1 2 Thus, M is Ricci parallel. By Theorem 4.1, (4.1) holds. Due to (1.2) and Itˆo’s formula, by taking derivative of (4.1) with respect to t at t = 0, we obtain 1 Hess∆f (v1, v2) (6.7) 2 1 = (∆Hess )(v , v ) ic(v , Hess#(v ))+tr ( , v )v , Hess#( ) , f C∞(M). 2 f 1 2 − R 1 f 2 hR · 2 1 f · i ∈ 0 25 Now, letting u, v TxM with u = v = 1 and u, v = 0, and combining (6.7) with (6.3) for v = v = v, and∈ f C∞(M| )| with| | Hess (x)=h u i u, we arrive at 1 2 ∈ 0 f ⊗ 1 k = k( u 2 v 2 u, v 2)= (Hess ∆Hess )(v, v) | | | | −h i 2 ∆f − f = ic(v, Hess#(v))+tr ( , v)v, Hess#( ) = ec(u, v). −R f hR · f · i S Therefore, ec = k. S (c) (4) ec = k. By (6.6), (6.4) is equivalent to ⇒ S 1 (6.8) ∆Hess Hess , Hess = k d Hess 2 (∆f)2 . 2 f − ∆f f HS k f kHS − ∞ We first prove that (6.8) implies ic = 0. Let f C0 (M) with Hessf (x) = 0. For any ∞ ∇R ∈ u TxM, let h C0 (M) with Hessh(x)= u u. Applying (6.8) to fs := f + sh,s 0, we obtain∈ at point ∈x that ⊗ ≥
s s2 (∆Hess Hess )(u,u)+ (∆Hess Hess )(u,u) 2 f − ∆f 2 h − ∆h = k ds2 Hess 2 s2(∆h)2 , s> 0. k hkHS − −1 Multiplying by s and letting s 0, we arrive at (∆Hessf Hess∆f )(u,u)=0. As shown above, this implies ic = 0. → − ∇R Next, we prove ec = k. Since ic = 0, (6.7) holds. For x TxM and u, v TxM with u = v = 1 andS u, v = 0, take∇Rf C∞(M) such that ∈ ∈ | | | | h i ∈ 0 Hess (x)= u v + v u. f ⊗ ⊗ Then at point x, Hess#( )= u, v + v, u. f · h ·i h ·i So, by (6.7) and (6.8) we obtain
2kd = k d Hess 2 (∆f)2 (x) k f kHS − 1 (6.9) = ∆Hess Hess , Hess (x) = (∆Hess Hess )(u, v) 2h f − ∆f f iHS f − ∆f =2 ic(u, Hess#(v)) 2tr ( , v)u, Hess#( ) =2 ic(u,u)+2 ec(u, v). R f − hR · f · i R S Letting v be orthonormal and orthogonal to u, replacing v by v and sum over i { 1}1≤i≤d−1 i leads to 2kd(d 1) = 2(d 1) ic(u,u)+2 ic(u,u)=2d ic(u,u). − − R R R Thus, ic(u,u)=(d 1)k, and (6.9) implies ec(u, v)= k. R − S
26 7 Identifications of Einstein manifolds
Theorem 7.1. For any constant K R, the following statements are equivalent each other: ∈ (1) M is an Einstein manifold with ic = K. R (2) < , and for any x M, t 0, f C (M) and v , v T M, kRk∞ ∞ ∈ ≥ ∈ 0 1 2 ∈ x t −Kt −Ks (7.1) HessP f (v1, v2) e PtHessf (v1, v2)= e Ps HessP − f (v1, v2)ds. t − Z R t s 0 (3) < , and kRk∞ ∞ eKt e2(K−kRk∞)t − Hess 2 P f 2 eKt P f 2 2 K k Ptf kHS ≤ t|∇ | − |∇ t | kRk∞ − e(2kRk∞−K)t 1 − P Hess 2 , f C2(M), t 0. ≤ 2(d 1) K tk f kHS ∈ b ≥ − kRk∞ − (4) There exists h : [0, ) M [0, ) with lim h(t, )=0 such that ∞ × → ∞ t→0 · P f 2 eKt P f 2 h(t, ) Hess 2 + P Hess 2 , t 0, f C∞(M). t|∇ | − |∇ t | ≤ · k Ptf kHS tk f kHS ≥ ∈ 0 (5) < , and kRk∞ ∞ t Ks 2(K−kRk∞)s Kt 2 e e 2 2 e 1 2 HessP f − ds Ptf (Ptf) − Ptf k t kHS Z 2 K ≤ − − K |∇ | 0 kRk∞ − t (2kRk∞−K)t−Ks 2(kRk∞−K)s 2 e e 2 Pt Hessf − ds, f C (M), t 0. ≤ k kHS Z 2 K ∈ b ≥ 0 kRk∞ − (6) < , and kRk∞ ∞ t 2(kRk∞−K)s −Ks −Kt 2 e e ) 2 2 1 e 2 Pt Hessf − ds Ptf (Ptf) − Pt f − k kHS Z 2 K ≤ − − K |∇ | 0 kRk∞ − t 2(kRk∞−K)s −Ks 2 2(K−kRk∞)t e e 2 HessP f e − ds, f C (M), t 0. ≤ −k t kHS Z 2 K ∈ b ≥ 0 kRk∞ − (7) There exists h˜ : [0, ) M [0, ) with lim t−1h˜(t, )=0 such that ∞ × → ∞ t→0 · Kt −Kt 2 2 e 1 2 2 2 1 e 2 min Ptf (Ptf) − Ptf , Ptf (Ptf) − Pt f − − K |∇ | − − K |∇ |
h˜(t, ) Hess 2 + P Hess 2 , t 0, f C∞(M). ≤ · k Ptf kHS tk f kHS ≥ ∈ 0 (8) For any f C∞(M), ∈ 0 1 Hess ∆Hess =( Hess ) KHess . 2 ∆f − f R f − f Proof. Obviously, (3) implies (4), each of (5) and (6) implies (7). According to Theorem 4.1 for ic = K, (1) implies (2). Moreover, by taking derivative of (7.1) with respect to t at R t = 0, we obtain (8). So, it suffices to prove that (1) implies (3); (3) implies (5) and (6); and each of (4), (7) and (8) implies (1).
27 (a) (1) (3). By the Bochner-Weitzenb¨ock formula and using ic = K, we obtain ⇒ R 1 ∆ P f 2(X ) P f, ∆P f (X ) 2 |∇ t−s | s − h∇ t−s ∇ t−s i s 2 = ic( Pt−sf, Pt−sf)(Xs)+ HessPt−sf HS(Xs) R ∇ 2 ∇ k 2 k = K P f (X )+ Hess − (X ). |∇ t−s | s k Pt sf kHS s Then (4.8) implies 2 2 2 2 d P f (X )= K P f + Hess − (X )ds +2 P f (X ), Φ dB |∇ t−s | s |∇ t−s | k Pt sf kHS s h∇|∇ t−s | s s si for s [0, t]. Thus, ∈ t 2 Kt 2 K(t−s) 2 (7.2) Pt f e Pt f = e Ps HessPt−sf HSds. |∇ | − |∇ | Z0 k k Since by (4.3) 2 2(kRk∞−K)(t−s) 2 Hess − e P Hess , k Pt sf kHS ≤ t−sk f kHS it follows from (7.2) that t 2 Kt 2 K(t−s)+2(kRk∞−K)(t−s) 2 Pt f e Pt f e Pt Hessf HSds |∇ | − |∇ | ≤ Z0 k k e(2kRk∞−K)t 1 = − P Hess 2. 2 K tk f k kRk∞ − So, the second inequality in (3) holds. Similarly, (4.3) implies 2 −2(kRk∞−K)s 2 −2(kRk∞−K)s 2 P Hess − e Hess − = e Hess , sk Pt sf kHS ≥ k PsPt sf kHS k Ptf kHS the first inequality in (3) also follows from (7.2).
(b) (3) (5) and (6). By (3) and (4.3) we have ⇒ eKs e2(K−kRk∞)s − Hess 2 P P f 2 eKs P f 2 2 K k Ptf kHS ≤ s|∇ t−s | − |∇ t | kRk∞ − (2kRk∞−K)s e 1 2 − P Hess − ≤ 2 K sk Pt sf kHS kRk∞ − e((2kRk∞−K)t−K(t−s) e2(kRk∞−K)(t−s) − P Hess 2 . ≤ 2 K tk f kHS kRk∞ − This together with (4.9) ensures (5). Similarly, (4.3) and (3) imply e−Ks(e(2kRk∞−K)s 1) − P Hess 2 e−KsP f 2 P P f 2 2 K tk f kHS ≥ t|∇ | − t−s|∇ s | kRk∞ − 1 e(K−2kRk∞)s − P Hess 2 ≥ 2 K t−sk Psf kHS kRk∞ − e2(kRk∞−K)s e−Ks e2(K−kRk∞)t Hess 2 − , ≥ k Ptf kHS · 2 K kRk∞ − which together with (4.9) gives (6).
28 (c) (4) (1). For v T M with v = 1, take f C∞(M) such that ⇒ ∈ x | | ∈ 0 f(x)= v, Hess (x)=0. ∇ f
Then (4.11) holds. Moreover, since HessPtf (x) is smooth in t, Hessf (x) = 0 implies Hess (x) c(x)t, t [0, 1] k Ptf k ≤ ∈ for some constant c(x) > 0. Combining this with (4.11), (5.5) and (4), we obtain P f 2(x) eKt P f 2(x) 0 = lim t|∇ | − |∇ t | t↓0 t Kt 1 e 2 = lim ic(v, v)+ − Pt f (x) = ic(v, v) K. t↓0 R t |∇ | R − Therefore, ic(v, v) = K v 2 holds for all v T M. By the symmetry of ic and g, this is equivalent toR ic = K. | | ∈ R R (d) (7) (1). Let v and f be in (c). In the spirit of (4.11) and using (4.9), we have ⇒ Kt P f 2 (P f)2 e −1 P f 2 2 lim t − t − K |∇ t | t↓0 t2 t 2 2 Ks 2 = lim Ps Pt−sf e Ptf (x)ds = ic(v, v) K, t↓0 t2 Z |∇ | − |∇ | R − 0 and −Kt P f 2 (P f)2 1−e P f 2 2 lim t − t − K t|∇ | t↓0 t2 t 2 2 −Ks 2 = lim Ps Pt−sf e Pt f (x)ds = K ic(v, v). t↓0 t2 Z |∇ | − |∇ | − R 0 Thus, multiplying the inequality in (7) by t−2 and letting t 0, we prove ic(v, v) K =0. That is, (1) holds. → R −
∞ (e) (8) (1). For any v1, v2 TxM, take f C0 (M) such that f(x)= v1, Hessf (x) = 0. According⇒ to Theorem 5.1, (8)∈ implies ∈ ∇ ( ic)(v , v )=0. ∇v2 R 1 1 So, M is Ricci parallel, and as shown in the proof of Theorem 4.1 that (6.7) holds. Taking v = v = v for v T M with v = 1, and letting f C∞(M) such that Hess (x)= v v, 1 2 ∈ x | | ∈ 0 f ⊗ (6.7) implies 1 Hess ∆Hess (v, v)= ic(v, v)+tr ( , v)v, Hess#( ) = ic(v, v). 2 ∆f − f −R hR · f · i −R Combining this with (8) we obtain ic(v, v)= KHess (v, v)= K. −R − f − So, (1) holds.
29 8 Identifications of Ricci Parallel manifolds
Theorem 8.1. The following assertions are equivalent each other: (1) M is a Ricci parallel manifold. (2) < , and (4.1) holds for any x M, t 0, f C (M), v , v T M. kRk∞ ∞ ∈ ≥ ∈ 0 1 2 ∈ x (3) < , and for any constant K R with ic K, t 0, f C2(M), kRk∞ ∞ ∈ R ≥ ≥ ∈ b ic (1 e−Kt) (8.1) Hess P Hess kR k∞ − + e(kRk∞−K)t e−Kt P Hess . k Ptf − t f k ≤ K − tk f k
1 (4) There exists a function h : [0, ) M [0, ) with lim t− 2 h(t, )=0 such that ∞ × → ∞ t↓0 · (8.2) Hess P Hess h(t, ) P Hess + Hess , t 0, f C∞(M). k Ptf − t f k ≤ · tk f k k Ptf k ≥ ∈ 0 (5) For any f C∞(M) and x M, ∈ 0 ∈ Hess ∆Hess (v , v )=2 Hess (v , v ) 2 ic(v , Hess#(v )), v , v T M. ∆f − f 1 2 R f 1 2 − R 1 f 2 1 2 ∈ x (6) For any x M and f C∞(M) with Hess (x)=0, ∈ ∈ 0 f (∆Hess )(v , v ) = Hess (v , v ), v , v T M. f 1 2 ∆f 1 2 1 2 ∈ x Proof. The equivalence of (1) and (6) follows from Theorem 5.1, (1) implying (2) is included in Theorem 4.1, (5) follows from (2) by taking derivative of (4.1) with respect to t at t = 0, and it is obvious that (3) implies (4) while (6) follows from (5). So, it remains to prove that (1) implies (3), and (4) implies (1).
(a) (1) implies (3). Let M be Ricci parallel with ic K. By (1.2) we have R ≥ K (8.3) W (v) e− 2 s v , v T M, | s | ≤ | | ∈ x and
2 −1 −1 2 d Wt(v) tv = d Φt Wt(v) Φ0 v | −k | | − | K = W (v) v, ic#(W (v)) dt W (v) v ic e− 2 t v . h t −kt R t i ≤| t −kt |·kR k∞ | | So, t − K t ic K ic (1 e 2 ) ∞ − 2 s ∞ Wt(v) v kR k e ds = kR k − , v 1. | − | ≤ 2 Z0 K | | ≤ Thus, for v , v = 1, | 1| | 2| Hess (W (v ), W (v )) Hess ( v , v ) f t 1 t 2 − f kt 1 kt 2 Hess W (v ) W (v ) v ) + W (v ) v (8.4) ≤k f k | t 1 |·| t 2 −kt 2 | | t 1 −kt 1| −Kt K K ic ∞ ic ∞(1 e ) Hess e− 2 t +1 1 e− 2 t kR k = Hess kR k − . ≤k f k − K k f k K 30 Combining (4.1) with (3.6) for ζ = , (4.2), (8.3) and (8.4), we obtain ∞ Hess P Hess k Ptf − t f k ic (1 e−Kt) sup Hess (v , v ) EHess (W (v ), W (v )) + kR k∞ − P Hess ≤ Ptf 1 2 − f t 1 t 2 K tk f k |v1|,|v2|≤1 ic −Kt t ∞(1 e ) −Ks E kR k − Pt Hessf + ∞ e HessPt−sf (Xs)ds ≤ K k k kRk Z0 k k −Kt t ic ∞(1 e ) −Ks kR k − Pt Hessf + ∞ e Ps HessPt−sf ds ≤ K k k kRk Z0 k k −Kt t ic ∞(1 e ) −Ks+(kRk∞−K)(t−s) Pt Hessf kR k − + ∞ e ds ≤ k k K kRk Z0 −Kt ic ∞(1 e ) (kRk∞−K)t −Kt = kR k − + e e Pt Hessf . K − k k So, (8.1) holds.
∞ (b) (4) (1). For any x M and v1, v2 TxM, let f C0 (M) such that f(x) = v1 and Hess ⇒(x) = 0. Since Hess∈ (x) = 0 and Hess∈ (x) is smooth∈ in t 0, ∇ f f Ptf ≥ Hess (x) ct, t [0, 1] k Ptf k ≤ ∈ holds for some constant c> 0. Combining this with Theorem 5.1, (8.2) and (5.5), we obtain
PtHessf HessPtf (v1, v2) ( v2 ic)(v1, v1) = lim | − | ∇ R t↓0 t
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