The Lie Algebra of Local Killing Fields

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1 The Open Mathematics Journal, 011, 4, The Lie Algebra of Local Killing Fiels Open Access Richar Atkins * Department of Mathematics, Trinity Western University, 7600 Glover Roa, Langley, BC, VY 1Y1, Canaa Abstract:We present an algebraic proceure that fins the Lie algebra of the local Killing fiels of a smooth metric. In particular, we etermine the number of inepenent local Killing fiels about a given point on the manifol. As an application, we provie a local classification of the types of surfaces that amit the various possible Lie algebras of local Killing fiels, in terms of the Gaussian curvature. Keywors: Lie algebra, local Killing fiel, erive flag, curvature, locally symmetric space. 1. INTRODUCTION Killing fiels escribe the infinitesimal isometries of a metric an as such play a significant role in ifferential geometry an general relativity. In this paper we present an algebraic metho that fins the Lie algebra of the local Killing fiels of a smooth metric g. In particular, we etermine the number of inepenent local Killing fiels of g about any given point. In the section following, we ientify the local Killing fiels of a metric with local parallel sections of an associate vector bunle W, enowe with a connection. An examination of the form of the curvature of leas to a characterization of spaces of constant curvature by means of a system of linear equations. In Section 3 we investigate the Lie algebra structure of Killing fiels. Section 4 inclues an overview of the proceure evelope in [1]. Therein the bunle generate by the local parallel sections of W is foun by calculating a erive flag of subsets of W. The number of inepenent Killing fiels of g about a point x M is then equal to the imension of the fibre W x over x of the terminal subset of the erive flag. Associate to W x is a Lie algebra canonically isomorphic to the Lie algebra K x of local Killing fiels about x. The metho is illustrate by proviing a short proof of a classical theorem that gives a necessary conition for a space to be locally symmetric, expresse by the vanishing of a set of quaratic homogeneous polynomials in the curvature. Section 5 consiers the erive flag for surfaces. We obtain a complete classification of the local Riemannian metrics corresponing to the various possible kins of Lie algebra K x. Furthermore, we show that if a regular surface with non-constant curvature possesses a Killing fiel then the integral curves of the curvature vector fiel are geoesic paths. *Aress corresponence to this author at the Department of Mathematics, Trinity Western University, 7600 Glover Roa, Langley, BC, VY 1Y1 Canaa; Tel: ; Fax: ; richar.atkins@twu.ca. KILLING FIELDS AND CONSTANT CURVATURE We associate to Killing fiels parallel sections of a suitable vector bunle in the manner put forwar by Kostant []. The utility of such a framework is two-fol: first, it permits us to apply algebraic techniques aapte to fining the subbunle generate by local parallel sections. Secon, it enables a purely algebraic escription of the Lie bracket of two Killing fiels, avoiing the explicit appearance of erivatives. Let g be a metric on a ifferentiable manifol M of imension n ; g is assume to be pseuo-riemannian of signature (p,q) unless otherwise state. K is a Killing fiel of g if an only if, K a;b + K b;a = 0 (1) where the semi-colon inicates covariant ifferentiation with respect to the Levi-Civita connection of g. It is straightforwar to verify that, K a;bc = R abc K () for Killing fiels K, where R abc is the Riemann curvature tensor of g, efine accoring to, A c;ba A c;ab = R abc A The summation convention shall be use throughout. Let W be the Whitney sum W := T * M " T * M. A local section of W has the form X = K + L, where K = K a x a is a local section of T * M an L = L ab x a x b is a local section of T * M. Define a connection on W by, i X = (K a;i " L ai )x a c + (L ab;i " R abi K c )x a # x b (3) For an open subset U M, let K U enote the local Killing fiels K :U T * M an let P U enote the local parallel sections X :U W ; the subscript U shall be omitte when U = M. Define the map U : K U " P U by, / Bentham Open

2 6 The Open Mathematics Journal, 011, Volume 4 Richar Atkins U (K a ) := K a + K a;b (4) It is clear that the image of U oes, in fact, lie in PU. The inverse U : PU " KU of U is the projection of W onto T * M : U (K a + Lab ) := K a. This establishes a vector space isomorphism, PU KU (5) Consier a vector space V with a non-egenerate, symmetric bilinear form h. Let B = Babc be a covariant 4 tensor on V satisfying the following relations common to a Riemann curvature tensor: Expresse in terms of inices, M has constant curvature (for n 3 ) if an only if for all L " T * M, Rsjkl Ls i + Riskl Ls j + Rijsl Ls k + Rijks Ls l = 0 (9) It is evient from the lemma that the proposition oes not hol for n =. If has constant curvature then g Rijkl = 0 (" il" jk # " ik" jl ) with respect to an orthonormal (6) frame, where 0 is a constant. Substitution of this expression into the left han sie of (9) gives zero for all skew-symmetric L = Lab. an let T = Tab be an n -tensor on V with n. The erivation B T is the (n + ) -tensor efine by, Suppose that (9) hols for all L " T * M. We shall work in an orthonormal frame X1,..., X n for g ; this will Babc = Bcab = Bbac = Babc B Tabc := Bsbc T sa + Basc T sb + Babs T sc + BabcsT s (7) Inices are raise by h. Lemma 1. If V is -imensional then B L = 0 for all L " V *. It shall be convenient to work in an orthonormal basis of i V in which h =iag(1, ), where i = ±1. Then L j = i Lij. Owing to the symmetries (6), there are effectively two cases to consier. a = b = 1 case: B Labc = (B1c + B1c )L1 = 0 (ii) Rsjil Lsi + Risil Lsj + Rijsl Lsi + Rijis Lsl = 0 (10) Put Lrs := rl sj " rj sl into (10) to obtain Rijij = Rilil. It follows that for any two pairs of istinct inices i j an (i) i allow us to eal with lowere inices throughout: L j = Lij. Let i = k, j an l be three istinct inices in (9). This gives, a b, Rijij = Rabab. Thus, Rijij = (x) for i " j (11) where is some function on M. Next, let i = k an j = l be two istinct inices in (9). This gives: Rijis Lsj + Rijsj Lsi = 0 a = c = 1, b = = case: (1) B Labc = B1 L1 + 1 B111 L1 + B1 L1 + 1 B111 L1 = 0 Let m be any inex istinct from i an j an put Lrs := rm sj " rj sm into (1). We obtain, Rijim = 0 Applying the Bianchi ientities, F(i, j) := [i j ] of takes the form, F(i, j)(x) = (Rijkl;s K s + R Lijkl )x k x l the curvature (8) where X = K a x a + Lab x a x b [3]. In the sequel, it shall be convenient to view the curvature F as a map F :W " T * M # W given by w F(, )(w). F is compose of two pieces: a K -part an an L -part. The K -part provies a escription of locally symmetric spaces: g is locally symmetric if an only if T * M ker F. The L -part, on the other han, provies a characterization of metrics of constant sectional curvature by means of a system of homogeneous linear equations. Proposition. Let g be Riemannian an n 3. Then M is a space of constant curvature if an only if, RL =0 for all L " T M * for i, j an m istinct (13) Consier a pair Y1,Y of orthonormal vectors in Tx M. If X1, X span the same plane as Y1,Y at x then R(Y1,Y,Y1,Y ) = (x), by (11). If Y1,Y span a plane orthogonal to X1, X then we may as well suppose X 3 = Y1 an X 4 = Y, whence R(Y1,Y,Y1,Y ) = (x), from (11) again. The last possibility is that Y1,Y an X1, X span planes that intersect through a line, which for the purpose of calculating sectional curvature we may take to be generate by X1 = Y1, by means of appropriate rotations of the pairs X1, X an Y1,Y within the respective planes they span. We may suppose, furthermore, that X 3 is the normalize component of Y orthogonal to X ; thus, Y = ax + bx 3, where a + b = 1. From (11) an (13) this gives,

3 The Lie Algebra of Local Killing Fiels The Open Mathematics Journal, 011, Volume 4 7 R(Y 1,Y,Y 1,Y ) = R(X 1,aX + bx 3, X 1,aX + bx 3 ) = a R 11 + b R 1313 = (x) Therefore g has constant curvature at each point x M. By Schur's Theorem, g has constant curvature. 3. THE LIE ALGEBRA STRUCTURE OF K U Let V be an n-imensional vector space equippe with a non-egenerate, symmetric bilinear form h, of signature (p,q), an let B = B abc be a covariant 4-tensor on V satisfying the usual algebraic relations of a Riemann curvature: B abc = B bac (14) B abc = B abc (15) B abc + B acb + B abc = 0, an afortiori (16) B abc = B cab By virtue of (14) an (15) we may efine a skewsymmetric, bilinear bracket operation on V * " V * by, [K a + L ab, K a + L ab ] : = L ab K b " L ab b + L a c c L cb " L a L cb + B abc K c (17) where inices are raise an lowere with h. If a subspace W of V * " V * is close with respect to the bracket an satisfies the Jacobi ientity then we enote the associate Lie algebra by A(W, B,h). Lemma 3. Let W be a subspace of V * " V *, close with respect to the bracket operation. The Jacobi ientity hols on W if an only if for all X = K + L, X = K + L an X = K + L in W, where K, K, K "V * an L, L, L "# V *, B L abc c + B L abc c K + B L abc K c = 0 (18) Let K, K, K "V * an L, L, L "# V *. There are four cases to consier. (i) K K " K "" case. We have, [K,[ K, K ]] = [K, B abc c ] = B abc K b c Therefore, [K,[ K, K ]]+[ K,[ K, K]]+[ K,[K, K ]] = (B abc + B acb + B abc )K b c = 0 (19) by equation (16). (ii) K K " L case. First, [K,[ K, L]] = [K, L ab b ] = B abc K c L s s Also, [L,[K, K ]] = [L, B abc K c ] = B asc K c L s " Las b B sbc K c Combining these with (15) an the fact that L = L ab is skew-symmetric, we obtain, [K,[ K, L]]+[ K,[L, K]]+[L,[K, K ]] = B L abc K c (0) (iii) K L L " case. Observe that, an, [K,[L, L ]] = [K, L L " L L ] = L LK " L L K [L,[ L, K]] = "[L, L K] = L L K Using these equations gives, [K,[L, L ]]+[L,[ L, K]]+[ L,[K, L]] = 0 (1) (iv) L L " L "" case. It is elementary to verify that, [L,[ L, L ]]+[ L,[ L, L]]+[ L,[L, L ]] = 0 () After applying (19)-(), simplifies to, [X,[ X, X ]]+[ X,[ X, X]]+[ X,[X, X ]] B L abc c + B L abc c K + B L abc K c Proposition 4. (i) If V is -imensional then any subspace W of V * " V *, close with respect to the bracket, efines a Lie algebra A(W, B,h). (ii) If V is arbitrary an B = 0 then V * " V * efines a Lie algebra A(V * " V *, B = 0,h). (i) Follows from Lemmas 1 an 3. (ii) is immeiate. For local Killing fiels K, K "K U, the Lie bracket K := [K, K ] is, K a = L ab K b " L ab b (3) where we have written L ab := K a;b an L ab := K a;b. Furthermore, L ab := K a;b is given by, L ab = L a c L cb " L ac L cb + R abc K c (4) Defining a bracket on P U by, [K + L, K + L ] : = L ab K b " L ab b + L a c L cb " L ac L cb + R abc K c (5) gives an isomorphism U : K U " P U of Lie algebras: U ([K, K "]) = [ U (K), U ( K ")] (6)

4 8 The Open Mathematics Journal, 011, Volume 4 Richar Atkins For x U, efine the subspace WU,x of Wx by, WU,x := {w Wx : w = X(x), for some X PU } (7) Since parallel sections of a vector bunle are etermine by their value at a single point, PU an WU,x are isomorphic as vector spaces via the restriction map rx : PU WU,x, given by rx (X) := X(x). Comparing (17) an (5), we have, in fact, efine a leveling map S as follows. For any subset V of W satisfying P1 let S(V ) be the subset of V consisting of all elements v for which there exists a smooth local section s :U M " V W such that v = s( (v)). Then S(V ) satisfies both P1 an P. We may now escribe the construction of the maximal, of W. Let, flat subset W a Lie algebra isomorphism rx : PU A(WU,x, Rx, gx ). Com- V (0) posing this with U characterizes the Lie algebra of KU. W Lemma 5. V rx U : KU " A(WU,x, Rx, gx ) (i+1) := {w W F(, )(w) = 0} (i ) := S(V ) := ker " (8) W (i ) where, as before, F :TM TM W " W is the curvature tensor of. This gives a sequence, is an isomorphism of Lie algebras. In orer to fin the Lie algebra of the local Killing fiels of g about the point x it remains to calculate WU,x for a sufficiently small neighbourhoo U accomplishe in the following section. (i ) of x. This is 4. PARALLEL FIELDS AND LOCALLY SYMMETRIC SPACES We begin by briefly reviewing the metho from [1]. This escribes an algebraic proceure for etermining the number of inepenent local parallel sections of a smooth vector bunle :W " M with a connection. Since the existence theory is base upon the Frobenius Theorem, smooth ata are require. Let W be a subset of W satisfying the following two properties: P1: the fibre of W over each x M is a linear subspace of the fibre of W over x, an, P: W is level in the sense that each element w W " is containe in the image of a local smooth section of W, efine in some neighbourhoo of (w) in M. Let X be a local section of W. The covariant erivative by, of X is a local section of W T * M. Define (X) := " " #(X) where :W " T * M # (W / W $ ) " T * M enotes the natural projection taken fibrewise. If f is any ifferentiable function with the same omain as X then, ( fx) = f (X) efines a map, This means that W " : W " # (W / W " ) $ T * M which is linear on each fibre of W. The kernel of W " is a subset of W, which satisfies property P1 but not necessarily property P. In orer to carry out the above constructions to ker W ", as we i to W, the non-level points in ker W " must be remove. To this en we W W (0) W (1) W (k ) of subsets of W. For some k N, W (l ) = W (k ) for all l k. = W (k ), with projection :W ""M. Define W We say that the connection is regular at x M if " there exists a neighbourhoo U of x such that "1 (U ) # W x shall enote the fibre of W is a vector bunle over U. W over x M. Lemma 6. Let be a connection on the smooth vector bunle :W " M. If X :U M " W is a local parallel section then the. image of X lies in W (i) (ii) Suppose that is regular at x M. Then for every x there exists a local parallel section X :U M " W w W with X(x) = w. We may now escribe the Lie algebra of local Killing fiels about a point x. Theorem 7. Let g be a smooth metric on a manifol M with associate connection on W = T * M " T * M, which is assume to be regular at x M. Then g has x inepenent local Killing fiels in a sufficiently imw small neighbourhoo U of x M. Moreover, the Lie algebra of Killing fiels on U is canonically isomorphic to x, R,g ). the Lie algebra A(W x x By Lemma 6 there exists a sufficiently small open x. The theorem neighbourhoo U of x such that WU,x = W now follows from Lemma 5. As an illustration, we provie a short algebraic proof of the classical theorem that locally symmetric spaces satisfy, RR=0 [4]. (9)

5 The Lie Algebra of Local Killing Fiels The Open Mathematics Journal, 011, Volume 4. Lemma 8. If M is locally symmetric then T * M W 9 As observe above, a Riemannian manifol with im M 3 has constant curvature if an only if R L = 0 for all (Proposition ). Furthermore, all manifols of imension n = 1 or satisfy R L = 0 for L " T * M (cf. Lemma 1). Since spaces of constant curvature are locally symmetric the curvature F vanishes for such manifols. It is not ifficult to see that the converse hols (for the imensional case use the canonical form of the Riemann curvature: Rijkl = c(gil g jk gik g jl ), where c is the Gaussian L"T *M Suppose M is locally symmetric an let x M. Then there is an open neighbourhoo U of x such that the space of Killing fiels on U, whose covariant erivative vanishes at x, has imension n. By the isomorphism given in Lemma 5, Tx* M WU,x. From Lemma 6 (i) we have x an so T * M W x. W W U,x x Let T = Ta be an n -tensor with n 1. Define R T to be the (n + ) -tensor obtaine by contracting the rightmost inex of R with the leftmost inex of T : R Tabc := Rabc s Ts (30) Let p := T * M, the cotangent space of M an let p (1) be curvature). Consequently, =W. M is a space of constant curvature if an only if W (34) Employing Theorem 7, we obtain as a corollary the familiar result that a Riemannian manifol possesses the 1 maximal possible number n(n + 1) of inepenent local Killing fiels if an only if it is a space of constant curvature. efine as the set of all elements v T * M satisfying, R Rv = 0 (31) Define t to be the set of all L " T M such that, * (3) RL =0 We shall assume that t has constant rank. Proof of (9): 5. CLASSIFICATION FOR RIEMANNIAN SURFACES In this final section we shall etermine which Riemannian surfaces correspon to the various types of Lie algebra K x. In the process, a necessary conition for a Riemannian surface to possess a Killing fiel is obtaine. First, we recall the situation involving the maximal number of local Killing fiels: Let us calculate the erive flag of W supposing M to be locally symmetric. From (8), W (0) = V (0) := ker F = p t. Let M be a Riemannian surface. The following are equivalent: Let X = K + L be a local section of W (0), where K an L are local sections of p an t, respectively. By efinition, (i) im K x = 3 for all x M. X(x) Vx(1) if an only if i X(x) "Wx(0), for all i. This is equivalent to, (ii) M has constant Gaussian curvature c. (iii) M is locally symmetric. (*) : Lab;(i ) Rab(i )c K c "t at x. Taking the covariant erivative of R L = 0 gives R L;(i ) = 0. Thus Lab;(i ) is a local section of t. This means that (*) hols if an only if R K x lies in t. This is the case precisely when K p (1) at x, an so V (1) = p (1) t. By Lemma 8, we must have V (1) = T * M t, whence it follows W (1) = V (1) = W (0). Therefore that an p (1) = T * M RR=0. We have also shown that for M locally symmetric, = pt then W = W (0) = p t. Conversely, if W T * M ker F, from which we conclue that M is locally symmetric. The terminal subbunle of the erive flag therefore characterizes locally symmetric spaces: = p t. M is a locally symmetric space if an only if W (33) In particular, the erive flag computes the local canonical ecomposition. In this case, (a) if c = 0 then K x is isomorphic to the Lie algebra of " s SO() ; (b) if c > 0 then K x sl " ; (c) if c < 0 then K x su. The equivalence of (i)-(iii) follows from Lemma 1, (34) an the observation below (34). Assume that these conitions hol. c = 0 correspons, locally, to flat Eucliean space, for which K x is isomorphic to the Lie algebra of the semiirect prouct of translations an rotations. Suppose c 0 an let X an Y be orthogonal vectors in Tx* M with norm, X = Y = 1 c Define H " Tx* M by, [X,Y ] := H Then, [H, X] = sg(c)y an [H,Y ] = sg(c)x

6 10 The Open Mathematics Journal, 011, Volume 4 Richar Atkins where sg(c) enotes the sign of c. Appealing to Lemma 5, this ientifies K x with sl for c > 0 an with su for c < 0 [5]. Next, we calculate the erive flag for W assuming that the surface is regular: W (i ) = V (i ) has constant rank for all i. The elements K T * M satisfying Rijkl ;s K s are those for which c,s K s = 0. Therefore, by Lemma 1, W (0) = ker c " # T M * (35) where c enotes the vector fiel c,a. The case c = 0 has been hanle above. Suppose therefore that im ker c = 1 ; that is, c is non-vanishing. Then W (0) is a rank-two fibre bunle over M. Let X = K + L be a local section of W (0), where K is a local section of ker c an L is a local section of T * M. X(x) Wx(1) is equivalent to i X(x) "Wx(0) for all i, by the efinition of the erive flag. Since T * M " W (0), X(x) Wx(1) if an only if, K a;(i ) La(i ) "ker #c (36) At x. Taking the covariant erivative of c,s K s = 0 gives the equation c,a K a;b = c;ab K a. Substituting this into (36) etermines W (1) as the subset of all K + L W (0) satisfying, c;ab K a + Lab c,a = 0 (37) By contracting (37) with K an c, it is evient that W (1) consists of the zero elements in W along with the solutions of, c;ab K a K b + Lab c,a K b = 0 (38) c;ab K a c,b = 0 (39) where 0 K "ker #c an L " T * M. Equation (39) has a solution 0 K "ker #c if an only if, c;ab c,b = fca (40) for some f. (40) may be written in terms of ifferential forms as, c D"c c = 0 (41) where D enotes covariant ifferentiation. This leas to a necessary conition for the existence of a Killing fiel on a Riemannian surface. Theorem 9. If a regular Riemannian surface possesses a Killing fiel then, c D"c c = 0 By Theorem 7, if a regular Riemannian surface has a has rank at least one. Since W W (1), Killing fiel then W equation (39) must have a non-trivial solution K ker "c at each x M. The theorem now follows from the fact that (39) is equivalent to (41). Corollary 10. Let M be a regular Riemannian surface with non-constant curvature. If M possesses a Killing fiel then the integral curves of c are geoesic paths. By a geoesic path we mean a curve that is a geoesic when appropriately parameterize. Equation (41) is equivalent to Dc c = f c, which implies that integral curves of the non-vanishing vector fiel c may be parametrize so as to give geoesics of M. An example woul be the puncture paraboloi z = x + y ; (x, y) (0, 0), with the inuce metric from its embeing into 3-imensional Eucliean space. The integral curves of c are escribe by the geoesic paths " (t) = (tcos",tsin",t ), up to reparametrization. Now let us return to calculating W (1). If c D"c c = 0 on the surface then the non-zero elements of W (1) are the solutions to (38) for which 0 K "ker #c. For any choice of non-trivial K ker "c, (38) uniquely etermines an element L = L(K ) " T * M. Therefore W (1), in this case, is a rank one vector bunle over M. If, on the other han, c D"c c # 0 on M then W (1) is the zero bunle an there o not exist any local Killing fiels. As a consequence, K x cannot be -imensional; this may also be seen irectly by consiering the Lie bracket operation. Henceforth we shall assume that c D"c c = 0. To fin W (), let X = K + L be a local section of W (1). By efinition, X(x) Wx() if an only if i X(x) "Wx(1) for all i. Owing to (37), this is equivalent to c;ab K (i a) + Lab(i ) c,a = 0 (4) Where, K a(i ) := K a;(i ) " La(i ) Lab(i ) := Lab;(i ) " Rab(i )c K c (Note that from the escription of W (1) containe in (36) it follows that K a(i ) + Lab(i ) "W (0) ). Taking the covariant erivative of (37) gives, a,a a ;a c;ab K ;(i ) + Lab;(i ) c = c;ab(i ) K Lab c (i ) (43) Substituting (43) into (4) efines W () as the subset of all K + L W (1) such that, c;abc K a + Lab c ;a c + Lac c ;a b + Rabc c,a K = 0 (44)

7 The Lie Algebra of Local Killing Fiels The Open Mathematics Journal, 011, Volume 4 11 If this has only the trivial solution then W = W () is the zero bunle. Otherwise, W = W () = W (1) has rank one. Theorem 11. Let M be a regular Riemannian surface. Then im K x = 1 for all x M if an only if, (i) c 0, (ii) c D "c c = 0, an (iii) equation (44) hols for all K + L W (1). Conitions (i)-(iii) are equivalent to rank W = 1. The result now follows from Theorem 7. We summarize the iscussion in this section with the following corollary. Corollary 1. For regular Riemannian surfaces, K x may be one of five possible Lie algebras. It is isomorphic to either sl, su or the Lie algebra of " s SO() when the Gaussian curvature is constant an positive, negative or zero, respectively. K x is the 1-imensional Lie algebra when the conitions of Theorem 11 are met. Otherwise, there o not exist any local Killing fiels an K x is trivial. REFERENCES [1] Atkins R. Existence of Parallel Sections of a Vector Bunle. JGP 011; 61: [] Kostant B. Holonomy an the Lie algebra of infinitesimal motions of a Riemannian manifol. Trans Amer Math Soc 1955; 80. [3] Console S, Olmos C. Curvature invariants, Killing vector fiels, connections an cohomogeneity. Proc Am Math Soc 009; 137(3): [4] Helgason S. Differential geometry, lie groups, an symmetric spaces. Grauate stuies in mathematics American Mathematical Society 001. [5] Fulton W, Harris J. Representation theory. Grauate texts in mathematics. Springer Receive: November 04, 009 Revise: December 8, 010 Accepte: February 8, 011 Richar Atkins; Licensee Bentham Open. This is an open access article license uner the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricte, non-commercial use, istribution an reprouction in any meium, provie the work is properly cite.