Killing Fields Generated By Multiple Solutions to the Fischer-Marsden Equation

Transcription

1 Killing Field Generated By ultiple Solution to the Ficher-arden Equation Paul Cernea and Daniel Guan October 13, 2014 Abtract In the proce of finding Eintein metric in dimenion n 3, we can earch metric critical for the calar curvature among fixed-volume metric of contant calar curvature on a cloed oriented manifold. Thi lead to a ytem of PDE which we call the Ficher-arden Equation, after a conjecture concerning thi ytem for calar function, involving the linearization of the calar curvature. The Ficher-arden conjecture aid that if the equation admit a olution, the underlying Riemannian manifold i Eintein. Counter-example are known by O. Kobayahi and J. Lafontaine. However, almot all the counterexample are homogeneou. ultiple olution to thi ytem yield Killing vector field. We how that the dimenion of the olution pace W can be at mot n + 1, with equality implying that, g i a phere with contant ectional curvature. oreover, we how that the identity component of the iometry group ha a factor SOW. We alo how that geometrie admitting Ficher-arden olution are cloed under product with Eintein manifold after a recaling. Therefore, we obtain a lot of non-homogeneou counter-example to the Ficher-arden conjecture. We then prove that all the homogeneou manifold with a olution are in thi cae. Furthermore, we alo proved that a related Bee conjecture i true for the compact homogeneou manifold. In emory of Profeor S. Kobayahi Acknowledgment and Dedication Thank to Profeor ichael T. Anderon for making the firt author aware of problem involving the linearization of the calar curvature. The firt author alo wihe to thank Joeph alkoun for encouragement and converation about mathematic. The econd author became a graduate tudent of Profeor S. Kobayahi in the Fall 1990 after he finihed [2] with Profeor Dorfmeiter, in which he ued the Ricci curvature fibration reult from [11]. He eventually claified thoe compact homogeneou complex manifold with invariant volume tudied by [11] in [7]. In 2012, Guan ued [11] to give everal different proof of the recent claification of Profeor Haegawa on the compact homogeneou locally conformal Kähler manifold. any year before Guan came to United State, he had copie of the famou book [10] and never imagined that he himelf would become one of the tudent of the famou differential geometer S. Kobayahi. In thoe three year in Berkeley, Guan had written and publhed a few paper under Profeor Kobayahi. [5] wa Guan referee report for one of Profeor Gang Tian paper and wa publihed by Profeor Kobayahi. When Guan finihed [6] in the Spring 1992, Profeor Kobayahi helpped him mooth the language. That eventually led to the olution of cohomogeneity one compact Kähler Eintein manifold in, for example, [9]. That olution i cloely related to the holomorphic vector bundle cae preented in [12], ee [8]. The econd author dedicate thi work to Profeor Kobayahi. One can ee the power of the Lie group action. Univerity of California, Riveride, Department of Computer Science and Engineering, pcern001@ucr.edu Univerity of California, Riveride, Department of athematic, zguan@math.ucr.edu 1

2 The econd author alo like to take thi paper to honor. Lu, Lingzi Dorothy, who died in the Boton Bombing April 2013 and got the firt perfect core in Guan ordinary differential equation cla ATH 46 during her final in Univerity of California at Riveride in Fall Introduction and Summary of Reult Let be a cloed, connected, orientable manifold of dimenion n 3. Conider the calar curvature a a function on the pace S of Riemannian metric of fixed unit volume and contant calar curvature. Define the Laplacian a the trace of the Heian = g ij i j. Eigenvalue of the Laplacian are necearily nonnegative contant λ 0 for which there exit function u C, not identically zero, uch that u + λu = 0 1 Beware that in Bee [1], for intance, the oppoite ign convention i ued for. From Koio [14], we can conclude that, for any g S, if / i not a poitive eigenvalue of the Laplacian, then, for any ymmetric bilinear 2-tenor h uch that Lh := i j h ij h ij g ij h ij R ij = 0 and h ij g ij dµ = 0 2 we can find a one-parameter family gt in S with g 0 = h. Thu, for generic g S, the et of thee h can be thought of a the tangent pace of S. L i in fact the linearization of the calar curvature, o that g+th+ot t 2 = Lh 3 t=0 Following [1] page 128, uppoe g i a metric with / not a poitive eigenvalue of the Laplacian o = 0 i allowed. Define a metric g S to be critical for the Eintein-Hilbert action Eg = gdµ if, given any one-parameter family gt in S with derivative g 0 = h a above, we have d dteg0 = 0. Then g i critical in thi ene if and only if there exit ome function f C uch that L f ij := i j f fg ij fr ij = R ij n g ij 4 where L denote the L 2 adjoint of L. For completene, we outline a proof in the appendix. Now, taking the trace of Equation 4, we obtain f + f = 0 5 o that, ince / i not a poitive eigenvalue, we mut have f a contant in fact, zero and g mut be an Eintein manifold R ij = /ng ij. Bee [1] 4.48 goe further and ak, what if / i in the pectrum? If g obey Equation 4 and o i formally critical, mut g be Eintein? If g i not Eintein, f cannot be a contant. In thi work, we chooe to focu on what happen if there are multiple olution f 1 and f 2 to 4. Indeed, ince f i an eigenfunction of the Laplacian, we can write u := 1 + f and rewrite 4 a the critical metric equation i j u = ur ij u 1 g ij 6 n If u = 1 + f 1 and v = 1 + f 2 are olution to the above, then their difference x = f 1 f 2 olve the linear equation i j x = x R ij g ij 7 2

3 and x i an eigenfunction of the Laplacian with eigenvalue /. The Ficher-arden Conjecture aked whether g that atify 7 are Eintein. Counterexample to that have been found ee, for intance, Kobayahi [13] and Lafontaine [15]. We notice that almot all the known example are homogeneou and the dimenion of the olution pace of 7 i at leat 2. We will how that in general any product metric of the form S m N where N i Eintein yield a counterexample. We will call an x atifying 7 a Ficher-arden olution. If u and v are olution to the critical metric equation 6, then udv vdu i a conformal Killing field. Even nicer, if x and y are Ficher-arden olution, then Y = x y y x 8 i a Killing field a oberved in Lafontaine [16] where the ituation in dimenion n = 3 i tudied. We how that uch a Killing field atifie the equation R iljk Y l = R ij Y k R ik Y j g ijy k g ik Y j 9 where R ijkl are component of the Riemann curvature tenor uch that R ij = R kikj, and RicY = ρy 10 for ome mooth function ρ defined where Y 0 depending on g but not on choice of Y. Furthermore, if w i any Ficher-arden olution, then o i dwy = Y i i w. There i a contant β < 0 uch that, if x and y are L 2 -orthonormal Ficher-arden olution, then x 2 + β x 2 = y 2 + β y 2 = ix i y xy = ρ We ue thi to prove that the pace of Ficher-arden olution ha dimenion le than or equal to n + 1 with equality only if, g ha contant curvature. In fact, we prove the tronger tatement Theorem A. Let W be the pace of Ficher-arden olution of 7, and I be the identity component of the iometry group of, g. Then I i locally SOW G 1 with a compact Lie group G 1 which i the kernel of the action of I on W. oreover, all the SOW orbit are either S dim W 1 or it fixed point. We then how Theorem B. If, g admit a Ficher-arden olution u of 7, then after a poible recaling it product with a poitive Eintein manifold alo ha u a a Ficher-arden olution of 7. Thu we exhibit many nonhomogeneou geometrie admitting olution of 7. In the homogeneou cae, we obtain a convere: Theorem C. If, g i cloed homogeneou manifold admitting a nontrivial Ficher-arden olution of 7, then we can write = S dimw 1 N with N a homogeneou Eintein manifold. We hall tudy the nonhomogeneou cae and equation of 6 from [1] in the near future. Here, we like to mention that if there i a olution for 6 on a non-eintein manifold on Eintein manifold 6 and 7 are eentially the ame, we can alway chooe u to be invariant under the iometry group and W Ru 1 i an invariant ubpace of eigenfunction ince any difference of two olution of 6 i a olution of 7. In particular, there i no homogeneou manifold for 6 except that i Eintein. Thi i becaue if, g i homogeneou and there i a invariant olution u of 6, it mut be a contant. The left ide of 6 i zero and the right ide of 6 implie that g i Eintein. That i, Theorem D. The Bee conjecture i true for compact homogeneou Riemannian manifold. Thi i poibly the firt reult for the Bee conjecture. Throughout thi paper, normal coordinate at conidered point are ued. See [10] part I page 148 for a reference. In ection 2 and thereafter, all Riemann metric are aumed to have contant calar curvature. 11 3

4 2. The Killing Field and the Induced ap Firt we remark that if u and v are olution to for ome contant α, then i j u = ur ij j u i v v i u = j u i v i u j v u α g ij 12 α u vg ij 13 i u j v v j u + j u i v v i u = 2α v ug ij 14 o that udv vdu i a conformal Killing field if α 0 and a Killing field if α = 0 ee [1] page 40 for baic propertie for Killing vector field. oreover, any two calar-function olution u and v to 12 differ by a Ficher-arden olution, o we henceforth retrict our attention to the equation i j x = x R ij g ij 15 and denote olution to the above by x and y, with the Killing field they generate by Y i = {x, y} i = x i y y i x 16 Our mot important tool will be the map f A Y f = Y i i f = dfy with f C. Thi induced map i kew-ymmetric with repect to the L 2 -inner product: ua Y vdµ + va Y udµ = Y i i uvdµ = i Y i uv dµ = 0 17 if Y i divergence-free. Thi i alo becaue the iometry group i compact and therefore, any finite-dimenion repreentation of it i kew-ymmetric. Furthermore, if Y i a Killing vector field, A Y take eigenfunction of the Laplacian to eigenfunction of the Laplacian with the ame eigenvalue. The induced map A Y alo take Ficher-arden olution to Ficher-arden olution ince Y i a Killing vector field. Thi i becaue, if ϕ i an iometry and u i a Ficher-arden olution, then ϕ u i a Ficher- arden olution ince the defining equation i in term of Riemannian invariant. Thu, given a Killing field Y and it one-parameter family of iometrie ϕ t, we have that ϕ t u are olution. Taking the partial derivative with repect to t, we have that Y i i u = A Y u i a olution. Henceforth, we write Au for A Y u if there i no poibility of confuion. Now we tudy the induced map in more depth. Propoition 1. Let Y = {x, y} be a Killing field generated by Ficher-arden olution x and y. Let A be the map induced by Y. Then Y 2 = xay yax, and A 2 x = β 2 x and A 2 y = β 2 y for ome contant β. Proof. We have Next Y 2 = x i y y i x Y i = xay yax 18 0 = 2Y i Y j i Y j = Y i i Y 2 = Y i i xay i yax + x i Ay y i Ax = xa 2 y ya 2 x 19 4

5 If x = 0 at a point, then y = 0 or A 2 x = 0 there. But if x = y = 0 at a point, then Y = 0 there o that A 2 x mut alo equal zero there. So the nodal vanihing et of x i contained in the nodal et of A 2 x. It follow from an obervation in Gichev [4] that the eigenfunction x and A 2 x mut be linearly dependent. Actually, what [4] aert i the following. Let u and v be eigenfunction correponding to the ame eigenvalue. If a nodal domain connected component of the complement of the nodal et of u i contained in that of v, then u = cv for c a contant. Now let N [u] denote the nodal et of u and uppoe N [v] N [u]. Then N [u] c N [v] c. If a connected component of N [u] c i not contained in a connected component of N [v] c, then the boundary of the latter, namely N [v], interect N [u] c. But that i impoible ince N [v] N [u]. Thu if one nodal et i contained in another, we can conclude that the eigenfunction are linearly dependent. We ee that x and y are eigenvector of A 2 with the ame eigenvalue. Since A i kew-ymmetric by 17, we have xa 2 xdµ = Ax 2 dµ 0 20 o we can write A 2 x = β 2 x and A 2 y = β 2 y for ome contant β. Thi alo follow from 19. Thi prove the propoition. Propoition 2. Special Form Of the Curvature. Let Y be a Killing field generated by Ficher-arden olution x and y. We have R iljk Y l = R ij Y k R ik Y j g ijy k g ik Y j 21 and R j i Y j = ρ Y Y i for ome function ρ Y defined where Y 0 for which dρ Y Y = 0. Proof. Let Y = {x, y}. Taking it covariant derivative, we have by 15 j Y k = j x k y xy R jk k x j y + xy g jk R jk g jk 22 j Y k = j x k y k x j y 23 Since Killing field Y atify i j Y k = R iljk Y l ee, for intance, Bee [1], 1.81, page 40, again taking the covariant derivative and applying 15 we obtain R iljk Y l = R ij g ij Y k R ik g ik Y j 24 Then 0 = R iljk Y i Y l Y k = R ij Y i Y 2 Y j Y 2 RcY, Y Y 2 Y j 25 o that R ij Y j = R k i Y k = ρ Y Y i for ome function ρ Y. Then by divergence-freene of Y, ymmetry of Ric, and kew-ymmetry of Y, dρ Y Y = Y i i ρ Y = i ρy Y i = i R ij Y j = 0 + R ij i Y j = 0 26 ince i a contant, by the econd Bianchi i R ij = 0 ee [1] page 120, 4.19 and the propoition follow. 5

6 3. The Ficher-arden Solution Space and the Iometry Group At thi point we ee, given Y = {x, y}, that x and y are pecial eigenfunction for A Y. However it i not clear how y and Ax are related. Let u explore thi now. Propoition 3. Let Ȳ and Y be two Killing field that are pointwie linearly dependent. Then they are linearly dependent a vector field. Proof. If they are pointwie linearly dependent, then Then where Y 0, Therefore, It follow that Ȳ i Y j = Y i Ȳ j Ȳ i = gȳ, Y Y 2 Y i = fy i 27 j Ȳ i = j fy i + f j Y i 28 i fy j = j fy i j fy j = 0 29 f 2 Y 2 = j fy j 2 = 0 30 everywhere, which i only poible if f i contant, and Ȳ and Y are linearly dependent. Propoition 4. Let Y = {x, y} be a Killing field generated by L 2 -orthonormal Ficher-arden olution x and y. Let A be the induced map. Then there i a contant β < 0 uch that Ax = βy Ay = βx 31 x 2 + β x 2 = y 2 + β y 2 = ix i y xy = ρ Y Proof. Since Ax i alo Ficher-arden, we know we can get another Killing field by 32 But Ȳ i = x i Ax Ax i x 33 So Thu i Ax = i x j x j y i y x 2 + x ρ Y Ȳ i = x i x j x j y x i y x 2 + x 2 ρ Y Ȳ i = Ax = x j x j y y x 2 34 [x 2 ρ Y Y i + xy yx R ij By Propoition 3 we ee that there i ome contant β uch that gij j x 35 Y i x i x j x j y + y i x x 2 36 x 2 ] Y i 37 6

7 x 2 ρ Y x 2 = β 38 We mut have β 0. Otherwie Ȳ = 0, and x i Ax = Ax i x. Contracting with Y now yield xa 2 x = Ax 2. Actually, β ha nothing to do with y and only depend on x. We could alway make it to be 1 by recaling a we hall ee later on on the unit phere. Integrating, we have Ax2 dµ = Ax2 dµ and o Ax = 0 and hence A 2 x = 0. But from Propoition 1, A 2 x/x = A 2 y/y o that alo A 2 y = 0. But then 0 = ya2 ydµ = Ay2 dµ o that Ay = 0. Since Y 2 = xay yax by 18, we have Y identically 0. But then dx/x = dy/y o that y = Cx, which contradict the orthogonality of x and y. So indeed β 0. oving forward, from Ȳ = βy we have x i Ax βy = Ax βy i x 39 Separating variable and olving the differential equation, there i a contant C uch that Integrating, 0 = Ax = βy + Cx 40 xaxdµ = β xydµ + C x 2 dµ 41 Thu C = 0 ince xydµ by the aumption of L2 -orthogonality. So Ax = βy. Then Ay = β 2 / βx by Propoition 1. We conclude that β 0, by reaoning imilar to that which implied β 0. Now write c = β/ β and x = x c and ỹ = y/ c. Then we have A x = βỹ and Aỹ = β x. Integrating ỹa x + xaỹ = β ỹ 2 x 2 42 we ee that x2 dµ = ỹ2 dµ, which i the ame a aying β 2 = β 2 becaue x and y are orthonormal by aumption. Since we have not yet fixed the ign of β, we will fix β = β. By conidering the ytem of equation for Ax and Ay: and 38, we deduce x i x i y y x 2 = βy x y 2 y i x i y = βx 43 x 2 + β x 2 = y 2 + β y 2 = ix i y xy = ρ Y Conider the firt equality in 44, and examine a point where x = 0 and y 0 uch a point mut exit, or ele x and y are linearly dependent. We ee that β < 0, and thi finihe the proof. Propoition 5. The contant β < 0 and ρ Y do not depend on the choice of Killing field Y generated by Ficher-arden olution x and y, and we write imply ρ. oreover, if A i the induced map and u i a Ficher-arden olution orthogonal to x and y, we have Au = 0. Proof. Let x, y, and z be orthonormal Ficher-arden olution. Conider Y = {x, y} and Z = {x, z} and U = {y, z}. Let β, β, and β be their repective contant. Since Ric i ymmetric, we have But at a point where Y 0 and Z 0 and x 0, we have 44 ρ Y ρ Z Y i Z i = R ij R ij Y i Z j = 0 45 ρ Y ρ Z = x 2 + β x 2 x 2 + β x 2 = β β x

8 by the firt and lat expreion in 44. Thu if ρ Y = ρ Z we have β = β. Otherwie Y i Z i = 0 on an open dene ubet, o that in fact Y i Z i = 0 everywhere. But that i Dividing by x 2 yz we get x 2 i y i z xy i x i z xz i x i y + yz x 2 = 0 47 i y i z yz ix i z xz ix i y xy Subtituting expreion involving the β contant from 44 into 48 yield + x 2 x 2 = 0 48 x 2 + β + β x 2 = y 2 + β y 2 49 By conidering 44 and 49 together, we ee that if x = 0 and y 0, then β + β = x 2 = β. So in fact, β = 0, a contradiction. So actually ρ Y = ρ Z and β = β. If w i yet another olution, we have that ρ and β are the ame for {x, y} and {x, w} = { w, x} and { w, z} = {z, w}. Now conider a Ficher-arden olution u orthogonal to x and y if any uch exit. Then Au = x i y y i x i u = xyu yxu ρ = 0 50 which prove the propoition. With all the machinery in place, we can etablih: Theorem 1. Let W be the Ficher-arden olution pace, and I the identity component of the iometry group of, g. Then I i locally iomorphic to SOW G 1, where G 1 i the kernel of the repreentation of I on W. In particular, W ha dimenion at mot n + 1, with equality implying that, g i the round phere. oreover, all the SOW orbit are either phere or it fixed point. Proof. We continue our argument in the proof of Propoition 5. With Y = {x, y}, Z = {x, z}, and U = {y, z}, we calculate [Y, Z] j = Y i i Z j Z i i Y j = βu j 51 Therefore, x y Y i a Lie algebra iomorphim from W W = ow to it image in the Killing field. One could alo ue the following elementary argument for a proof of the upper bound on dimenion: Let x i be an L 2 -orthonormal bai for the Ficher-arden olution pace. If there are at leat n + 1 of the x i, then chooe thee and conider {x i, x j }. Thee are nn + 1/2 Killing field. Let u how that they are linearly independent. To ee thi, uppoe to the contrary that 1 i<j n+1 α ij x i k x j x j k x i = 0 52 for contant α ij not all zero. In particular, uppoe that ome pecific α ij i nonzero. Relabelling indice if neceary, we can uppoe α Then 0 = 1 i<j n+1 n+1 α ij x i k x j x j k x i k x 1 = β α 1j x j 53 by Propoition 4 and 5. But the x j are linearly independent and β 0, o thi would mean α 1j = 0 for every j: a contradiction. Thu, g ha nn + 1/2 linearly independent Killing field, and o i maximally ymmetric. There cannot be more linearly independent Ficher-arden olution, or that would induce an j=1 8

9 even higher degree of ymmetry. Being maximally ymmetric,, g mut have contant curvature. Since, g i a cloed Eintein manifold, we have for ome Ficher-arden olution x: i j x + n xg ij = So that, g i iometric to the round phere by Obata Theorem [17] Theorem A. In general, the tangent pace of SOW orbit at a given point i a ubpace of the pace generated by x i /x i x j /x j and therefore, the dimenion i k 1. The orbit paed through that point i a phere or a point. 4. Example and the Homogeneou Cae Since the round phere i an example of a Riemannian manifold with nontrivial olution to the Ficher- arden equation, it may be aked if there are other. Lafontaine in [16] how that S 1 N i alo an example, where N i a urface of contant poitive curvature. We how that our tructure reult fit quite well with thee known example. For example, if = S 1 N, we could let β = 1 and x = co θ, then dx = in θdθ. The tandard metric give x 2 1 x = 1. Since S 1 i totally geodeic, we have ρ = R 2 11 = 0, we could let n 1 = 1 and R ii = 1 with i > 1. If n > 1, let x be one of the Euclidean coordinate for S n in R n+1, we can alo conider x = co θ and dx = in θdθ. Then, x generate geodeic. dx 2 = in 2 θ with the tandard metric for the unit phere. dx 2 1 x = 1. ρ =, = n and therefore, ρ 2 n 1 = n = 1 alo. For the coordinate x, y, we have dax + by 2 1 = ax + by 2 with a, b an unit vector. We then have dx, dy = xy a in 44. A a generalization, we exhibit the following example. Theorem 2. Product With Eintein anifold. If V, g i a manifold admitting a Ficher-arden olution u, and N, g i a cloed oriented Eintein manifold with Eintein contant c 2 > 0, then we can alway recale V o that u i a olution to the Ficher-arden equation on = V N with metric g = g + g. If V, g = S 1, dθ 2, we do not recale, but rather we et u to be cocθ or incθ. If there are multiple olution on V, g, the quantitie β and ρ on, g will be the ame a thoe on V, g. Converely, if, g i a product of V, g and N, g and admit a Ficher-arden olution u, then u mut be the pull-back of a olution on one of the factor, and the other factor mut be Eintein. Proof. If g and g are Riemannian metric, then the product metric g = g +g on V N ha Ricci curvature Ric = Ric + Ric. The calar curvature i = +. The Heian Ddfh of function f C V and h C N atifie Ddfh = hddf Ddfh = fddh Ddfh = df dh 55 for vector tangent to V, vector tangent to N, and the cae where one vector i tangent to V and the other to N, repectively. Let V have dimenion m. Recale V, g o that it calar curvature V atifie V = m 1c 2, where c 2 = /n m i the Eintein contant of N, g, and n = dim. Thu the calar curvature of, g atifie = V + n mc 2 = m 1 + n m c 2 = c 2 56 Let u be a Ficher-arden olution on V if m > 1, or if m = 1, chooe u to be cocθ or incθ. Working on, g, for vector tangent to N, we have i j u = 0 = R ij c 2 g ij u = R ij g ij u 57 9

10 ince the Eintein contant of N, g atifie c 2 = /n m. For vector tangent to V, we have, if m > 1, i j u = R ij u = R ij u 58 V m 1 g ij ince V /m 1 = c 2 = /. If m = 1, we have i j u = d2 u dθ 2 = 0 c2 u = R ij g ij g ij u 59 If one vector i tangent to V and the other i tangent to N, the cro-term vanih. In all cae we have i j u = R ij g ij u 60 Thu u i a olution on to the Ficher-arden equation. Now u 2 i the ame whether and g are taken with repect to V or with repect to. Alo, if x and y are two olution, x i y y i x i x will be the ame on V and, o that β will not have changed. Therefore ρ V V m 1 = x 2 + β x 2 = ρ Since V /m 1 = /, we alo have that ρ i invariant. For the convere, uppoe that, g i a product of V, g and N, g and admit a Ficher-arden olution u. The equation for the cro-term i u x i x j = 0 62 where i indice are tangent to V and j indice are tangent to N. Thi i becaue in uch a cae, the Chritoffel ymbol Γ k ij vanih, a do the term g ij and R ij. The only way thi equation can hold i if u = fh + f 1 + h 1, where f and f 1 are function on V, and h and h 1 are function on N. However, then the equation i i f j h + j f i h = 0, which implie 61 f 2 h 2 = i f i h 2 63 The only way that can hold i if one of the function, ay without lo of generality h, i contant ay equal to h 0 on a nonempty open et U On the complement of U, f mut be contant. But then h h 0 i an eigenfunction of the Laplacian on all of. Since it vanihe on a nonempty open et, we mut have h = h 0 on all of. So we can write u = f + h without lo of generality. Plugging thi u in for the equation with indice tangent to V and N, repectively, we ee that either f or h mut be contant. Without lo of generality, let h be contant, o that u = f. Now that we know u i the pull-back of a function on V, we mut have 0 = i j u = R ij g ij u 64 for i and j tangent to N. It follow that N, g i Eintein with poitive Eintein contant /. On the other hand, for ome contant τ, we have i ju = R ij τg ij u 65 for indice i and j tangent to V. Tracing give u u = τmu, where m i the dimenion of V. Taking the divergence yield j u = ju R ij j u = τ ju 66 10

11 whence we conclude that τ = /m 1. Thu u i a Ficher-arden olution on V, a deired. By looking at any example of the form S 1 N, we have x = cocθ and y = incθ, and o ρ = 0. Thu there are nontrivial example beide the phere where ρ i contant, and it become pertinent to prove Propoition 6. Let, g be a Riemannian manifold admitting k orthonormal Ficher-arden olution x 1, x 2,..., x k and uppoe ρ i a contant. If P i a homogeneou harmonic polynomial of degree α in k variable, then P x 1, x 2,... x k i an eigenfunction of the Laplacian with the correponding eigenvalue equal to α[α 1ρ α n 1 ]. Proof. If P i a real-valued function of the x i, we have P x = i = P x i + ρ x i i P x i x i + i,j i,j 2 P x i x j l x i l x j 67 2 P x i x j x i x j β P x 68 where we have ued the equation 44 to obtain the rightmot two term. Becaue P i homogeneou, thi implifie to P x = α P x + ρ αα 1P x β P x 69 Since P i harmonic and ρ i contant, we are left with [ P x = α α 1ρ α ] P x 70 a deired. Propoition 7. Let, g be a Riemannian manifold admitting k orthonormal Ficher-arden olution x 1, x 2,..., x k, then ρ i a contant if and only if x 1 i an eigenvector for the Ricci curvature operator with an eigenvalue ρ. In thi cae, x i generate geodeic, and either A all thoe x i have a mutual zero, or B x 2 1 +x x2 k i contant. In the latter cae B, we have ρ = k 2/n 1k 1, and x 1, x 2,..., x k i a harmonic map into S k 1. And moreover, i a product of S k 1 with a Eintein manifold. In the cae A, the common zero et of the olution are totally geodeic and all the geodeic generated by x i have the ame length. Proof. By taking the derivative of 44, we ee that ρ i contant if and only if x 1 i an eigenvector of the Ricci operator. Indeed, we have that ρ = 0 iff 2x 1 R ik for any i by applying 7 with C = ρ n 1. g ik k x 2Cx 1 i x = 0 x If x 1 i an eigenvector of the Ricci operator, then 1 x1 x 1 = 0 by applying 7 again. Therefore, the gradient generate geodeic, jut a in the cae of phere. Define r = x x x2 k. If the x i have no mutual zero, then r i mooth, and, for any σ R, we can conider r σ. We have r σ = σ σ 2r σ 4 x i x j k x i k x j + r σ 2 i,j i Uing again the equation in 44, we obtain x i 2 x2 i 71 11

12 r σ = σ σ 1 ρ r σ σβr σ 2 k 2 + σ 72 For k > 2, etting σ = 2 k, we ee that r σ i an eigenfunction of the Laplacian. But then it mut change ign on a cloed manifold, unle it i contant. So r i contant. Thi implie that the eigenfunction map harmonically into S k 1 ee [3]. oreover, we mut have σ 1ρ / = /, which implie ρ = k 2/k 1. For example, if k = n + 1, we get ρ = /n. If k = 2, we carry out a imilar analyi with log r, and obtain log r = 2ρ, which implie ρ = 0 by the Divergence Theorem. Thu we till have ρ = k 2/k 1. In the cae B, the fiber of the map are totally geodeic by [7] p. 52. And the orbit aociated with the Killing vector field Y ij obtained by 16 with x i and x j in the place of x and y are the ame a thoe of x i. Since x i generate geodeic, the orbit are totally geodeic alo. Therefore, = S k 1 N with N the fiber of the map. In the cae A, by [7] Propoition 3.1 in page 51, the common zero et of the olution i totally geodeic. Alo, ince x 1 generate geodeic, from 44 we have at a maximal point of x 1, ρ n 1 = a 2 < 0 and x = a 2 x 2 1. Therefore, x = a co t a. In the lat entence of the proof, it i very poible that: Let N a be maximal point et of x 1 and N a be the minimal point et of x 1. Then both of them are ubmanifold and there i one to one map between them introduced by the cloet point from the other ubmanifold. Thoe two point are connected by the geodeic generated by x 1. The ytem of thee geodeic generate a ubmanifold, which might be the required phere of radiu a. And then i the product S k 1 N with N = N a = N a. Then the quetion remain that: a Doe thi picture actually work out? b I ρ alway a contant? Corollary. If ρ i contant and the Ficher-arden olution pace ha multiplicity m, then any m 1 of the olution have a common zero. Proof. If the above formula for ρ hold, it uniquely define k. A ituation where ρ will be contant i when, g i homogeneou, a cae we can completely claify: Theorem 3. Let, g be cloed, homogeneou and admit a Ficher-arden olution. Then, g mut be of the form S m N where N i an Eintein manifold. Proof. We can apply Propoition 7 directly. But here we can offer another more group involved proof. The Ficher-arden defining equation i written in term of Riemannian invariant. Therefore, if W i the pace of Ficher-arden olution, then the iometry group G ha W a an invariant ubpace. Thu there i a Lie group homomorphim G SOW. Since G i compact, it i reductive, meaning that it Lie algebra can be written a a, where i emiimple and a i abelian. By the claification of imple Lie group, thi mean that we can write G i locally iomorphic to a finite covering G = SOW G. The firt factor mut be SOW ince it orbit are phere or fixed point and phere i imply connected. G act tranitively on, g by iometrie via γ m = π γ m, where π i the covering map G G. The iotropy group i π 1 of the iotropy ubgroup of G. G fix all the olution function. Therefore, the interection of G m with each SOW orbit i unique. Thi give a product for. Thu = S m N with the product metric. By Theorem 2, N mut be Eintein. 5. Appendix Here we prove the aertion that tenor h atifying 12

13 i j h ij h ij g ij h ij R ij = 0 and h ij g ij dµ = 0 73 can alway occur a firt derivative of one-parameter familie gt in S, provided / i not in the poitive pectrum of g0. To ee thi, conider that from Theorem 2.5 in Koio [14], we have that, given g0 S, if we have a mooth perturbation gt, which might run outide of S, we can alway write gt = ft gt 74 with f0 = 1 and 0 < f C, o that gt actually lie in S for t cloe enough to zero. It only remain to how that we have enough control to make any h ij a above the derivative of gt at zero. Indeed, let g ij = g ij 0 + th ij 75 Then g 0 ij = h ij + f 0g ij 0. Writing g ij = g ij 0 for brevity, what we now need to how i that f 0 = 0. What we know i that f 0g ij i in the kernel of L, that i, the Laplacian compoed with the linearization of the calar curvature. Thi i becaue, on one hand, h ij i in the kernel of L and, on the other hand, the linearization of i L at g ince the calar curvature i contant. Thu Lg ij 0 = 0. So f 0g ij atifie Thu [ i j f 0g ij f 0g ij g ij f 0g ij R ij] = 0 76 i j f 0g ij f 0g ij g ij f 0g ij R ij = c 77 for ome contant c, ince harmonic function are contant on a cloed manifold. But thi i jut f 0 c 1 + f 0 c 1 = 0 78 o that f 0 i equal to ome contant c 1. Thu g 0 ij = h ij + c 1 g ij. However, we know g ij h ij dµ = g ij g 0 ij dµ = 0 79 So c 1 = 0 and g 0 = h a deired. Now we how that, if g S with / not in the poitive pectrum of g, then g i critical for the Eintein-Hilbert action Eg = dµ if, and only if i j f fg ij fr ij = R ij n g ij 80 for ome function f C. For, indeed, the left-hand ide L f i the adjoint of the linearization of calar curvature applied to f, a can eaily be checked. Now, given a path gt in S with initial poition g and initial velocity h, we have d dt dµ = Suppoe now that LL u = 0. Then t=0 = R ij h ij dµ = L u 2 dµ = i j h ij h ij g ij h ij R ij dµ 81 R ij n g ij h ij dµ 82 ull udµ =

14 We have L Rc n g udµ = R ij n g ij L u ij dµ = 0 84 Now, it can eaily be checked that LL i elliptic. By the Fredholm alternative, we mut alway be able to olve LL f = L Rc n g. Thu we have a Hodge-type decompoition R ij n g ij = L f ij + v ij 85 where v i in the kernel of L. Suppoe thi equation can be olved with v = 0. Then, for any initial velocity h, d dµ = L f dt ij h ij dµ = flhdµ = 0 86 t=0 ince h i in the kernel of L by definition. So g i critical. Otherwie, uppoe that there exit a nonidentically-zero olution v. Then take that v to be the initial velocity. We have d dµ = v 2 dµ < 0 87 dt o that g cannot be critical. Thi finihe the proof. Reference t=0 [1] Bee, A. L., Eintein anifold, Springer-Verlag, [2] Dorfmeiter, J. & Guan, Z. D., Claification of Compact Homogeneou Peudo-Kähler anifold, Comm. ath. Helv., , [3] Eell, J., Lemaire, L., A Report On Harmonic ap, Bull. London ath. Soc. 10, R 82b:58033 [4] Gichev, V.., A Note On Common Zeroe of Laplace-Beltrami Eigenfunction, Annal of Global Analyi and Geom., , [5] Guan, Z. D., Stability of Hermitian Vector Bundle, A Quantitative Point of View, Intern. J. ath., , [6] Guan, Z. D., Exitence of Extremal etric on Almot Homogeneou anifold with Two End, Tran. AS , [7] Guan, D., Claification of Compact Complex Homogeneou Space with Invariant Volume, Tran. AS., , [8] Guan, D., Exitence of Extremal etric on Almot Homogeneou anifold of Cohomgeneity One III, Intern. J. ath., , [9] Guan, D., Type I Compact Almot Homogeneou anifold of Cohomogeneity One III, Pacific J. ath., , [10] Kobayahi, S. & Nomizu, Foundation of Differential Geometry I, II, John Wiley & Son Inc [11] Hano, J. & Kobayahi, S., A Fibering of A Cla of Homogeneou Complex anifold, Tran. AS ,

15 [12] Kobayahi, S., Differential Geometry of Complex Vector Bundle, Publication the athematical Society of Japan 15, Iwanami Shoten, Publiher and Princeton Univerity Pre [13] Kobayahi, O., A Differential Equation Ariing From Scalar Curvature Function, J. ath. Soc. Japan, Vol , [14] Koio, N., A Decompoition of the Space of Riemannian etric on a anifold, Oaka J. ath , [15] Lafontaine, J., Sur la geometrie d une generaliation de l equation differentielle d Obata, J. de ath. pure et appliquee , [16] Lafontaine, J., A Remark About Static Space Time, Journal of Geometry and Phyic Volume 59, Iue 1, 2009, [17] Obata,., Certain condition for a Riemannian manifoldto be iometric with a phere, Journal of ath. Soc. Japan ,