Tensor Structures on a Differentiable Manifold

Home , Differentiable manifold, Torsion tensor

Tensor structures on ~, differentiable manifold.

by R. S. CLARK and M. Bt¢UCKHm.~Im¢(a Southampton~ Inghilierra}

To Enrico Bompiani on his scientific Jubilee

Summary.. A tensor structure is e. class of equiralent G-structures and it is deflated by a special tensor field. Such fields are characterized by the e~vistence of a linear connection relative to which they have covariant derivative zero. Two ts~tsor structures may admit a co~nmon subordinate ~tructure. E:~aples of such subordinate stuctures are given and some cases, when one stucture is a Riemannian metric, are considered.

INTRODUCTION The theory of G-structures on an n-dimensional differentiable manifold was introduced in 1953 by S. S. CHERN and it has been studied more re- cently by D. BER~AnD. The most familial" example arises when G is the orthogonal group 0(n, R), since this structure always exists on a Rieman- nian manifold. But almost complex structures have been studied by many writers, including C. EHR~:SMAN~, A. LICHN]~ROWICZ, P. LIBE~MANN, B. ECK- )IAO:~ and A. FROLICHER. Almost product structures have been investigated by A. G. WAL~:ER. T. J. WILLMORE and G. L~.GI~AND, among others. The present authors have introduced almost tangent structures. A differentiable manifold M admitting any one of the structures just mentioned carries a certain tensor field whose components, relative to some covering of M by moving frames, are constant. Such a tensor field we shall call special. Conversely, any special tensor field on M gives rise to a class of equivalent G-structures and we call this a tensor structure on M. It is sometimes convenient to pick out one particular represeniative for a tensor structure and call it the normal representative. For example, the G-structures mentioned in the first paragraph are the normal representatives of the tensor structures which they define. A connection for a G-structure on M determines a linear connection on M and this, in turn, defines a connection for any equivalent G-structure. We therefore define a connection for a tensor structure to be a linear connection on M which determines a connection for each of its representative G-strut. tares. We show that a linear connection, with absolute derivation D, is a connection for the structure defined by a tensor J if, and only if D J-" O. A necessary and sufficient condition that any differentiable tensor field J is special is that some such linear connection exists. 124 R. ~.~. ~~!.~.~l< - 3]. lb~Kul~x:,II:l,' : 7'{'J/.~'o{".~;Ft{~'fJ~rc.~" o;t

If M admits two tensor structures, then it is possible that two representative G-structures may admit a common subordinate structure. This would also be a G-structure and the class of all equivalent structures would be called a subordinate structure for the two tensor structures The almost quaternionic structures, introduced by C. EHt~ES~IAt¢I~ and P. LIBER]~AIql~', are representatives of such a subordinate structure. Suppose that 21/ carries two tensor structures, with teasers ,/1 and J2, which have a common subordinate structu,'e. We show that a linear connection on M, with absolute derivation D, is a connection for this subordinate structure if, and only if DJ,- 0 and D J2-- O. A necessary and sufficient condition that any two differentiable tensor fields, J~ and .]2, define tensor structures having a common subordinate structure is that some such linear connection exists. More important examples of a subordinate structure arise when one of the tensor structures is defined by a Riemannian metric. A Riemannian metric is said to be related to a given tensor structure if the two structures have a common subordinate structure. "We have defined a normal representative for a Riemannian structure. If a normal representative has also been defined for the given tensor structure and if the two normal representatives have a common subordinate structure, then the metric is specially related to the given tensor structure. A He rmitian metric is the simplest case of such a Riemannian metric, but a number of other examples are considered.

CHAPTER ONE. m Tensor structures.

1.1. G-structures. Let M be a differentiable manifold of class c~ and dimension n. If iU! is any covering of M by allowable coordinate neighbourhoods, the changes of coordinates define a 1-cocycle

u n ~] --. L(n, n} on M. Any two such cocycles are cohomologous and they define the principal fibre bundle of frames ~L --" HL ~M, L(n, R)). A moving frame 0: V is a differentiable cross-section 0 of ~EL of class c,¢ over an open set V of M. It may be identified with an ordered set of independent vector fields [0~ .... , 0,,] on 17. If we have an open covering of M by moving frames then, on an overlap, 0-- 0 T and the differentiable functions

T: V n V-- L (n, R) form a cocycle which also defiues ~L. l{. S. ('IAL~K - M. I~I{['I,~IIEIMI.:R: T('usor strm.turc., on a d([fcrcJ~htbh., etc. 125

Let G be a closed subgroup of the general linear group L{n, R} and suppose that we have an open covering of M as before except that T has values in G, that is~ we have a 1-cocyele on M with values in G. This defines a subordinate structure HL (M, L, G, H~) for ~L. Such a subordinate structure is a G-structure on M. Its associated principal fibre bundle is

JfG -" ttG(M, G).

HG can be identified with a subspace of tt5 and its points are called the frames adapted to the given G-structure. In just the same way, the bundle jfr, of coframes on M admits a subordinate structure with group G. We have a principal fibre bundle ~G whose local differentiable cross-sections 0"----i0 ~, ..., 0"} are the moving coframes adapted to the given G-structure.

1.2. Tensor structures of type (r, s). Suppose tbat we have a tensor field J of type (r, s) on M whose com- portents ~'(i)relative~(iJ to some covering of M by moving frames ~0: U! are constant. Such a tensor field will be called special. It is necessarily differentiable. On any overlap U 0 U, the change 0--0T of these frames must satisfy the condition

• • • oath... T~ T~....

The set of all such matrices T forms a subgroup G of L{n, R). This subgroup is algebraic and consequently it is closed. The differentiable functions

T: Uv~U~G form a 1-cocycle on M with values in G and define a G-structure on M. Clearly, J has the fixed components JJ~ relative to every adapted frame. Any other convering of M by moving frames, relative to which J has the same components, defines a cocycle which determines the same structure. Suppose that we are given a covering of M by moving frames I0': U' I 7,(t) relative to which J has constant components ~, c~. Then there exists a non-singular matrix X such that

:j) "" Xhl X~.~... = J'~'~h.... Xj, Xj2 ...

Relative to the moving frames 10X: UI, J has constant components ~'o) and these frames define an equivalent structure {1) on M with group X -t GX. But, from the above, IOX:UI and 10':U'I defin,~ the same structure and so we have proved 126 R.S. (~LRAK - M. ]{RUK]~IEIMI,]R: TC#8o~" structures o~, di]fcrc~t,bl% ~'l¢..

THEORE~ 1. - A special tensor field o f type (r, s) on M defines a clas~ of equivalent G-structures on M. Such a class of equivalent G-structures, determined by a special tensor field of type {r, s), will be called a tensor structure of type (r, s) on M. Any G-structure of the class will be called a representative structure. It is sometimes convenient to pick out one particular G-structure and call it the normal representative. We can do this by specifying the components J of J relative to its adapted frames. In the following examples we always give the normal representative, unless otherwise stated.

1.2. - ExampLes of special tensor fields.

(a) A non-zero differentiable vector field J on M defines a structure of type (0, 1). For, choose a moving frame [01,... 0,,] for a neighbourhood V of any given point m of M. Suppose the corresponding components of J are B i and that B~(m) =~= 0 for some k of 1 .... n. Then Bk=~:0 in a neighbourhood U~ V of m and the vectors

J, 0~, ... 0"k,... 0,,

form a moving frame on U relative to which J has components

J-" ti, 0, .... 01

We can find a covering of M by such moving frames. (b) A differentiable quadratic form J of constant rank r on M defines a structure of type (2,0). For, using the process of completing the square, at any given point m of M we can find a frame relative to which the symmetric tensor J has components

J--diag. /1,... (s terms}, -1,... (r-s terms), 0,...

It can be shown that there is a neighbourhood of m in which the process is differentiable and in which the integer s is fixed. Consequently we have a moving frame relative to which J has constant components J. We can cover M by such moving frames. (e) A differentiable exterior 2-form J of constant rank 2r on M defines a structure of type (2,0). For, at any given point m of M we can find a frame ]~. S. (~LRAK - .~[. BRUKHEIMER : Tensor structures on a differentabt G etc. 127 relative to which J has components

t 0 ... (r terms), 0,... and again this is true for a moving frame in a neighbourhood of qn. Such a matrix J~ is congruent to the nxn matrix

0 0 -I~ ] J-- 0 0 0 I~ 0 0 so that these components for J define another representative structure. This latter will be taken to be the normal representative.

1.4. - Tensor structures of type (1. 1).

Suppose that J is a special tensor field of type (1, 1) and let one fixed set of components be the matrix J. Then the components of J relative to any frame will have matrix form T -~ JT, where T is some non-singular matrix. Consequently, necessary condiEons for J to be a special tensor field are that it has constant rank on M and that it satisfies an algebraic equation. Sowe- times these conditions are sufficient and hawe

TREORE)I 2 - A differentiable tensor field J of type (1, 1) on M is special if ii has constant rank and satisfies an equation J~= kI, where the components of I form the unit nxn matrix I and k is any real number. To prove this theorem, we first suppose that ), = 1. Let V be a neighbourhood of a point m of M which admits a moving frame 0. The corresponding components B of J depend differentiably on V. The solution of the equations

(I- B) ~c = 0 (/+B)x=O

form complementary subspaces of R '~ depending on V- for they are disjoint and any vector w of /~" can be expressed as

1 1 (I + B) x + ~ (I-- B) x.

If at m the first subspace has dimension s~ the second has dimension n- s. Since we can find a neighbourhood U~V of m in which the ranks of I-- B and I+ B do not decrease, they will have constant rank on U. Thus we 128 R.S. CLRAK - M. ]~RUKttEIMER: Tel~.~or structurcs ~m o d~fj'ere~tttlble, etc. have linearly independent differentiable solutions

on U. Consequently the matrix X= [x~, .... x,] is non-singular and it depends differentiably on U. Since

I, 0 ] X-1BX =J: 0 --I._, , the tensor J has constant components J relative to the moving frame 0X: U. We can cover .M by such moving frames and so the theorem is true when -" 1. The normal representative here is an al,~ost product structure. When ).--- 1, EHRESMANN [2] has shown that we can cover M with moving frames relative to which J has components

J: Ir r:~n

This normal representative is an almost complex structure. Suppose next that ).--0 and that J has rank r. Then, using the nota- tion of the ease i( -- l, the matrix B has constant rank r. Hence the equation B~----0 has a differentiable and linearly independent set of solutions w~ (a = 1,...n-r) in some neighbourhood U. Let el,..., e,, be the natural basis for R" and replace n--r of the ei by the x~ to obtain at m a new basis

el) ... er, ~I, ..., X~--r where the e~ have been rearranged if necessary. The vectors Bel, .... Bet are linearly independent for, if ),aBea -- O, then B ()a e~,) -- 0 and so ).~ e,, belongs to the space spanned by the x~ and all ).a are zero. Also el, ..., e~, Bel,..., Bet are independent and we may replace the above basis by

el,..., e,., Be1,..., Be~, yl,..., Y,,-2~ where the y's are solutions of Bx = O. Since this new set of vectors is a basis for~ ]~ at m, it is a basis in some neighbourhood U of m. Let X be the matrix with these vectors as columns, then OX is a new moving frame. The components of J relative to these new frames are [o o] J-- L 0

Again, we can cover M by such moving frames and so the theorem is proved R. S. CLARK - M. BRUCKIIEIMER: Tcn.qor struvturc.~ on a dif]ere~ttiable, e~v. 129

t 1 for l--0. It follows from the proof that r ~< 2n. When r--2n, then this normal representative is an almost tangent structure. (3) Finally, the case when ~ is any real number reduces at once to one of these three eases and so the theorem has been proved. The theorem can easily be extended to the case when J satisfies the general quadratic equation

J~ + 2bJ + cl = O

For, putting K -- J+ bl, then K 2 -- (b 2 -- c)I and consequently the tensor field /~ is special if it has constant rank on M. Bat if K has constant components K relative to some covering, then J has constant components K + bI relative to the same covering and so we have THEOREM 2'. - A differentiable tensor field Y of type (1, 1) on M which satisfies the equation

(1) J~ + 2bJ + cl = O. is special if Y +b l has constant rank on M. The equation J~ -- J -- 0 is a familiar example. Such a special tensor field J defines the same structure as the special tensor field K= 2J- I, for which /¢2 _. I. The conditions that the tensor field J has constant rank on M and sati. sties equation (1) are not sufficient for it to be special. In fact, if it satisfies (1), J necessarily has constant rank on M unless b--c--0. For, unless b'--c, J is special and if b 2-- c :# 0, then it is non-singular.

CHA1~ER Two - Connections for tensor structures.

2.I. - Structure connections. Suppose given on M a G-structure with principal bundle ~G. A connec. tiou on ~G is a connection for the G-structure. Its form to is defined on the adapted frames Hs and it has values in the LIE algebra ~ of G. o) can be extended to all frames HL

~o(0X) = X-l~o(0) X O~HG,XzL(n,R) and it then defines a linear connection on M. We use the natural matrix representation of the LIE algebras of LIn, R~ and its subgroup G and then the above composition is matrix multiplication.

AnnaIi di Matematica 17 130 R.S. CLAaK - M. BRUCKHEIhlER: TC~I.nO~".~trt~ctm'es o~ a di]fcre~,tiable, etc.

If X is fixed, 0X are the adapted frames for an equivalent structure with group G' --X-1GX. The restricted form o) He, has values in ~' and it defines a connection for this equivalent structure. We may, therefore~ define a com~ection for ~r tensor structure to be linear connection whose form, when restricted to the adapted frames, of any representative structure with group G, has values in ~. Any linear connection defines an absolute derivation and we can eharacterise structure connections by. TKEORE~ 3. - A necessary and sufficient condition that a linear connec. tion on M, with absolute derivation D, is a connection for a structure with tensor J is that D J --0. To prove that this condition is necessary, suppose that we are given a connection for such a tensor structure. Let ~G be the representative bundle with group G. Then J has constant components tJ for the adapted frames Ho. Consequently a It J} is a zero form on He; and, therefore, so is D(~J ). It follows that D J is zero on M. That the condition is sufficient is included in

TttEORE~ 4. - If J is any differenliable tensor field on M and there exists a linear connection such that DJ-" O, then Y defines a tensor structure for whivh the given connection is a structure connection. This theorem can be proved by using theorems on holonomy given by 5Tomzu [4] and LlCttNE~OWlCZ [5]. However, we give the following direct proof. The condition D J-'-0 implies that tJ is constant along every horizontal curve on the space HL. Suppose that J has components J relative to a given frame. By .parallel transport of this frame along some curve in M, we can find a frame at every point of M relative to which J has components J. We must show the existence of a differentiable cross-section of ~L On which t,/ -" J over some neighbourhood of any point mo of M. Let mo belong to a bundle neighbourhood U of ~L which admits a coordinate system (~1,..., a~") in which mo has coordinates {0,..., 0}. The points of Hr. above U can be represented .by points of UX L(n, R). Suppose that J has components J relative to a frame (too, zo). Given any other point ml, of U with coordinates (X1,...X"}, we define a unique frame 0~, zl) at ml by parallel transport of the frame (rno, zol along the curve with equation x ~-- X ~ -:. We then have a cross-section of ~r. over U on which tJ --J. We have to show that it is differentiable. Since the tangent vector to the curve has components X, z~, is the solution of the equation IL S. {'t.Am< - 31. l;~t'{'l

,,valuated f,,r v-~ 1. But the ,:,luathJr~ has soiution

z = e,'p 1-- ":',>tXil. z0 and consequently

z~ = ex, p { -- o~ {X} ). Zo.

Since Zo ~ ex, p{--toiOij, zo, it follows that the section 0u, z) over U is given by

z = exp {-- ~) {XI ). zo and so it is differentiable. Thus J is a special tensor field. It remains to show that the giwm (.onnection for J~l. is a structure connection, that is, it must d,fine a connection for' any representative bundle ~(; of the tensor structure. A necessary and sufficient condition for this is that any horizontal tangent vecmrh t(, IlL at any point h of H~. must be a tangent vector to H~. But h is tangent to some horizontal curve in HL and, since DJ ~0, td is constant along any such curw~. Since the curve passes through h, it lies entirely in H~ and so h is a tangent ve(.tor to H(;. This completes the proof of the theorem. As an application of this theorem, suppose that we are giw..n any linear cenmmtion on 21I defined by local coefficients l'~k r~dative to some moving frame. Denote the corresponding cevariant derivation by D> Given any differentiable tensor field J of type {1, t} such that j2___ )./{with ).q=0} then, relative to the connection with coefficients

the absolute derivative of J is zero. Consequently J d~;termines a tensor structure on M. This provides an alternative proof for Theorem 2 when ), 4= 0.

2.2. - The Chern invariant. Suppose that M admits a G-structure and let ,~ be the torsion tensor of some structure connection. The components t ,~ of '~ relative to any adapted frame are defined on He; and have values in P-~R"®R,h/~,.Denotingthe vectors of the natural basis for R '~ by el, then e{k = ei® e~A e k is a basis for P. Consider the subspace W of P of elements of the form

where Ap is a basis for 6 and the real numbers ~j;P are arbitrary. 132 R,, S. (~J,A~K - M. lll¢lZCKHt,:l~i~'l~: 7'(~tJ,~' .~truclllrc.~ (~Jl, a d,iJ'fere~tiablc; etc.

Clearly, l/l/ is independent of the choice of the basi~ for ~. Let Z be some complementary space and ~ be the natural projection of P onto Z. BERNARD has shown that ~ (t,~b)-- C is a function on HG with values in Z which is independent of the choice of connection [6]. C is the Chern invariant [7] as. sociated with the G-structure. Given a covering of 3/ by adapted moving coframes 0, it can be shown that C is defined locally by ~ (tdO). BER~AI~D has further shown that any contravariant vector 2-form dO on M is the torsion tensor of some structure connection if, and only if

t~ is a tensor on HG with values in P of type aG, where ~g--g®~g-1. W is invariant under ~G. If Z also is invariant, BERNARD has proved that the CHimer invariant corresponding to Z is itself the torsion tensor of some connection for the G-structure. This theory can be used when a manifold M admits a tensor structure by applying it to some representative G-structure. In practice, we shall apply it to the normal representative. For example, a non-zero differeutiable vector field on M defines a (0, 1) tensor structure. The group of its normal representative is the affine group of dimension n-1. In this case the subspace Wis'Pitself and consequently --0. Thus the GHERN invariant C is zero and also any contravariant vector 2-form on M is the torsion tensor of a structure connection. In particular, M always admits a torsion-free structure connection.

2.3. - The Nijenhuis tensor for structures of type (1, 1). Given any differentiable tensor field J of type (1, 11 on a manifold M, NI;~ENHUIS (8) has defined on M a contravariant vector 2-form N. If, relative to any frame, J has components ~ and if Dk denotes covariant derivation with respect to any torsion-free linear connection on M, then the corresponding components Nj~ of N are

In particular, we can define the tensor N on any manifold which admits a tensor structure of type (1, 1) with tensor Y. Let ~ be the components of $ relative to an adapted frame of some representative strncture and let t~ be the corresponding local coefficients of a torsion-free linear connection. Then

"-" -- ~)k.

Using this equation in the above expression for N~, it can be verified that R. S. CLARK - 3[. BRUCKItEIMER: TeJmor structures on a differentiable, e~c. 133 the Nijer~huis tensor assooiated with any tensor structure of type (1, 1) on a two-dimensional manifold is zero. When J satisfies the equation J 2__ )~1, we have the alternative expres. sion

1 I -- B~(DkBa -- DhB,)i.

It follows that the ~'~IJENlqUIS tensor satisfies the following relations

B anti h i i h B ar,-ri (2) ~* in 2V B1N~h "- 0 BhNjk "~" i.L* hk --- O.

These were given by FROLICHER [9] for the case k =- 1.

2.4. - Relation between the •ijenhuis tensor and the Chern invariant. In this section and in 2.5 we refer only to a (1, 1) structure whose tensor J satisfies the equation J" -- ),I and has maximum rank. We obtain a relation between its ~IJENI-IUIS tensor N and the CHER~ invariant C of its normal representative. $ Suppose that 0 is an adapted moving coframe for this representative and that

t d 0~ = ~ ~, ~ Oi^ Ok .

Then the corresponding local coefficients ~o~k of any torsion-free linear con. nection on M satisfy

to~k-- ik-- Tik and so, using the formulae of 2.3, the corresponding components of N are given by

ll t 8 ~ k i t t • t 8 ( (3) NCk----7~ diJOn',, ~'~k + d,(J~y,k + Jky#)~

We now suppose that ).--0 or- I, so that the normal representatives are almost tangent or almost complex structures respectively. It can then be shown that one complementary subspace Z of W in P is spanned by

e~-~ b-~ ~ i, j, k -- 1, ..., n 1 a, b, c= 1,..., r--~n. 134 R.S. CLARK - M. ]:~RUCKIIEIMEIt : TeJt,~ov str uvlure,s o~ a, diffcrvntiablc, etc.

The projection ~: P ~ Z is given by

where all the ~ are zero except possibly

t -a "-- a+~" _a+r Gb+r c-[-r ~-~-r c-~-r- X(~b. ~+~ ~ Uc b+r -- do ~

In this case the components J~ of J for the adapted frames are all zero except J~+~ = 8~, J~+~ = ),8~ and so, for these frames, relations i2) show that

~.Vb+r ¢~-r --- k L~bc -- ~ c+r

Consequently using these relations and then (3), we find that

-~b+r--~ c--{-r --- 4 57~+~ ¢+, --- ~ ), Yb+,- -a ~+,

and so it follows that, when X- 0 or -1,

(5) ~ (tN) + X~ (td01 = 0.

Suppose next that X- 1, so that the normal representative is an almost product structure. If J has trac,~ 2s-n, one complementary subspace Z of W is spanned by

bc a, b, C "- 1,...,S ~,~, "r =s + 1,...,n.

The projection ~ is given as above, except that now all the c are zero except possibly

C --" G~y, Obo -- be"

Since, in this case, the components J~ of J are all zero except J~--8~ and J~--~, it follows that If. S. CLARK - ~[. BRUCKHEIMER: Te,sor structures on a d$fJerent~able, etc. 135

Using these relations and then (3), we find that

'-- a t~ --

and so relation (5) also holds when ), = 1. This relation 15) is easily seen to be independent of our choice for the subspace Z. It leads to

TItEOI~E~ 5. - Suppose given on M a (1, 1) structure whose tensor J satisfies the equation j2 = kI and has max.imum rank. Then its _Nijenhuis tensor N and the Chern invariant C for its normal representative satisfy the relation

,8 fiN) + ),O = O.

It follows from 14) that, for almost tangent and almost complex structures, the NuE~nUIS tensor vanishes if, and onlyif the Cmml~i invariant vanishes. This is also true for an almost product structure. But any G-structure admits a torsion-free connection if, and only if its CHERI~I invariant vanishes, and so we have TI-IEORE~t 6. - A tl, 1~ structure, whose tensor J satis/ies the equation d~ -- )J and has maximum rank, admits a torsion-free connection if, and only if its Nijenhuis tensor vanishes. When ;.=4=0, it follows from Theorem 5 that such a structure admits a connection with torsion tensor

1 ),

In fact, given any torsion-free linear connection on M defined by local coefficients Fj~ relative to some moving frame and denoting the covariant derivation 1)y Dk. then such a connection is determined by local coefficients

+ j (D. - + , + D,, B ')t

When ).--0, it follows from Theorems 5 and 6 that such a structure admits a connection with torsion tensor proportional to N if, and only if N--0.

2.5. lnvariant subspaees.

When a subspace Z is invariant under (TG, the corresponding CHERI,I invariant is itself the torsion tensor of some connection for the G-structure. For the almost product stlucture, the subspace Z already defined in 2.4 is invariant. For the almost complex structure, such an invariant subspace is 136 R.S. CLARK - ~[. BRUCKrlEIMER: Ten,sot .~tr,ucture.~ o~, a d~fferentiable, etc. spanned by the two sets of vectors

eb+r e+r b c b c+r bJ-r c i b+~,r c-~4r b e b c}r b~-r c

In contrast with these two cases~ we have TI~EOREM 7. - For an almost tangent structure, there is no subspace of P complementary to W which is invariant under ~G. To prove this theorem, we remark that the sabspace W has basis

eb. e b ¢-~-r b-~r (~+r ~b c-Fr ~ ea ~-va+r , va+r and, as we have already indicated in 2.4, one complementary subspace Z is spanned by the vectors eb+~ c÷~. Any other such subspace Z' mast admit a basis of the form

f~+~ ~+~ __ ~.b+~ e+~ + w~b+~ ~+~ for some vectors w of W. Consider the particular matrix

g= [,°--I, L 1 belonging to G. The corresponding transformation zg of the basis for W is given by

I be bc be , • be be r, (ag) ea ~ ea --- ~ Ca+,., ((~g~ ea+r ~ ea+r "-- U

(6) (Cry) ~va ~ ~a+~ ) -- kVa -r" va+~ ) = ea + ea+~ b b (~g) e~_~ -- e., ~ = ~a+,..

Also, it can be shown that

(og) °~+~b+~ o+~ -- e.+~b+~ •o~ ___ (e°+~be + e.+~b-~r c + e°+~ b c+r ) and consequently

~,,+,. = (e,+,. + e,,+,. + ~o+; J + (~g~ ,~°-F~ -- w,,+,. •

Since W is invariant under aG, the right-hand side of this equation belongs to W. If Z' were invariant, then the left-hand side would belong to Z' and the right-hand side would be zero. But, from equations (6), no combination of the basis of W exists to give w~r satisfying this condition. R. S. CLARK - M. BalrcKrmi~rF,R: Ten.~or str,etures on a diyferentiable, ere. 137

CHAPTER THREE. - Subordinate structures.

3.1. - Definition of subordinate structures.

Two structures on M with groups G~ and G2 have a common subordinate structure with group G~NG2 if M admits a covering by moving frames which are adapted for both structules. Suppose that two tensor structures on M, with tensors J1 and J2, admit representatives which have a common subordinate structure J~. The class of all structures equivalent to ~ is said to be a common subordinate structure for the tensor structures. A necessary and sufficient condition for the existence of such a structure is that M admits a covering by moving frames relative to which the components of Jx and J2 are constant. For example, suppose that M admits tensor structures of type (1, I) defined by tensors J and /~ such that J~:K"= --1.

It can be shown that they have a common subordinate structure if

JK ~-KJ --0 and among its representatives is the almost quaternionic structure defined by HRES~ANN (101. Similar remarks apply to the ease when

J'~ -- -- K-' -- ~ I J l~ ~- I~ J --0 which leads to the almost quaternionic structure of the second kind defined by LIBERMA~ (11).

3.2. - Connections and subordinate structure~. Suppose that we are given structures on M, with tensors J~ and J~, which admit a common subordinate structure. We define a connection for this subordinate structure to be a linear connection which determines a connection for each of its representatives. Suppose that ~[t~is such a representative. The components of both tensors define constant functions t J1 and t J2 on H and so, as in the proof of Theo- rem 3, for any connection for this subordinate structure we have D J1 --D J2-- O. Conversely, suppose that J1 and J2 are differentiable tensor fields on Air and that there exists a linear connection such that DJ~--D J2--O. Then it follows, from Theorem 4, that J ~ and J 2 define tensor structures. These tensor structures always have a common subordinate structure. For, if the tensors have components J1 and J2 for one frame, we can find a covering of M by moving frames relative to which they have the same components. Also, as in

Annali di Matematica 18 l;J14 R.S. i<',l,Altl'; - .'~l. tJttll('i,;ltf,;t.'tlEi: :,T~"t#.<:¢J~'" .~'/t'##f'/t¢t'f'.~ 0t1 a differ~'~ttiob#% ele.

tNe pt.o~ff ~t' Til~.~r(~n 4, it l'~ll(~ws th;ll the given conn,,tilden is ;, e.nn~;eti~n for lhc snl,ordinal- sti'uetilre. Ttiiis W,~ h;tve proved

TIIF, f)REM 7. - Lc/ 31 admit two tellsor struclure,s,, with teasers J2 :tnll J 2, which have a ~:ommon subordinate structure. A linear connecliol~, with absolute derivation D, is a conneelio. [or the subordinate struclur, if, and only if DJ1 = D d,~ -- O. THEOREM S. - If ~1 admits different|able tel~,s'~,r [ields J a and J~ (tvtd a linear connection relative to which DJa ~ l)Jz :- O, then J, and Jz define tensor structures on M with a common subordinate structure. The given co~- nection is a connection for this subordinate structure.

3.3. - Special Riemannian metrics. More important examples of subordinate strnc'lures arise wh(m ene of tl.. tensor structures is d~,fil~ed by a Riemannian metric, and so is a structure, of type (2,0) with a llorma| representatiw, hay|rig group 0(~,ll). Su('h a m~,- tric is said to b~ re/.ated ~o a given tensor structure if the two structur~-s have a common subordinate s~ructuve. A related metric exists for a given tensor struo|ur~; if, ;~nd only if the latter admits a representative whose group G can be redu(~ed to G(30fn,]~}. A sufficient conditi(~,t t'ov this is timt the eoset space, G/G(5 0(n, l/,j is solid. Suppose that a normal representative has been defined foragiven tensor structure. A Riemannian metri(, is then said to be specially related to the lensor structure if lhelr normal representatives hav[~ a common subordinate structure. In what follows we g)ve some examples of special metrics. (a) - The structure defined by a non-zero different|able vector field J on M has its normal repr('~sentative determined by compon(,nts J--I1,0 ..... 0! for J. A Riemannian metric S i.s special if, and oldy if d has unit length relative to S. For if the metric is special, we can find a, covering of M by moving frames relative to which S, J have components l,J respectively, and so the condition is necessary. Conversely, if the condition is satisfied, we can find a neighbourhood 17 of ~ny point m of M admitting a moving frame 0, relative to which S has components I and the compt)nents B of Jsatisfy B'B = t. We can then construct an orthogonal matrix X, depending (lifferen- tiably on a neighbourhood UcV of m. having B' for its first row. Relative to the moving frame 0X on U, the tensors S,J have compollcnts I, J respectively and so the metric S is special. {b) - The (1, 1) structure defined by a different|able tensor field J, such that J-'= !, has its normal representative {an almost product structure) determined by components

J= __/~_~°J , 1~, S. CLARK - ~. BIIUCKItEIMER: Tt'll,~or ,~trtwt~rrcs oJ1 a di]fcrcntiable, etc. 139

A Riemanniart metric S is special if, and only if SJ -" (SJ)'. For, this~con. dition is clearly necessary. Suppose that it is satisfied. Let V be a neighbourhood of a point m of M admitting a moving frame 0 relative to which S has components L The corresponding components B of J depend differentiably on V and are such that B---B' and B2___ L As in the proof of Theorem 2, we find independent vectors xl,..., xs; xs+l, ..., x,, of R ~ depending differentiably on a neighbourhood U of m. In this case the two sets of vectors are orthogonal, since ifor example)

~'~ xn -" (Bx,) ' (--Bxn) -" --~'1 x~.

By the usual process, each set can be replaced by a mutually orthonormal set (which will still be di[ferentiable) and w,~ then have an orthogonal matrix X. Relative to the moving frame 0 X on U, the tensors S, J have components I, J respectively and so the metric S is special. Such metrics exist for any structure of this kind. In fact, if S is any Riemannian metric, then

J is a special metric. The vectors of the adapted frames of the almost product structure ge- nerate complementary distributions on M. Let S be a Riemannian metric with components S relative to these ~dapted frames. The distributions will be orthogonal if, and only if S has the form

that is, if and only if JS---SJ. Consequently, a Riemannian metric S is special for this structure if, and only if the complementary distributions are orthogonal relative to S. (c) - The {1, 1p structure defined by a differentiable tensor field J, such that Y~--MI, has an almost complex structure as its normal representative. Again using a modification of the method referred to in the proof of Theorem 2, it can be shown that a Riemannian metric S is special if, and only if SJ =- (S J)'. Such a metric is Hermitian relative to the almost complex structure. The metric S is also special for this present tensor structure.

(d) - The (1, 1) structure defined by a differentiable tensor field J, of 1 constant rank r--2n and such that J~ 0, has its normal representative 140 II. S. CLARK - M. BRUCKIIE1MER: TcJ*sor str,wturcs on a d.i]]ere,~tiable, etc.

(an almost tangent struct.ure) determined by components

J= [ooL 0. t

A Riemannian metric S is special if, and only if

HJ ~ JH--I where H-- 5-1(S J}'. For, this condition is clearly necessary. Suppose that it is satisfied and that 0 is an adapted moving frame of the almost tangent structure for a neighbourhood U of any given point m of M. Then the corresponding components

of S satisfy the relations $1--Ss, S~--0. But $I is positive definite and so there exists a differentiable matrix T in a neighbourhood VC U of m such that S, -- T' T. Relative to the moving frame

on V, S and J have components I and J respectively. Consequently the metric S is special. Special metrics exist for this structure, since the group G for the almost tangent structure is the set of all matrices of the form

and the corresponding coset space GIG N 0(n, R) is solid. (e) - Similar methods show that a Riemannian metric S is special for the tensor structure on M defined by a non-singular differentiable quadratic form J if, and only if JS 1j "-"S and for the structure defined by a non-singular differentiable exterior 2-form ? if, and only if JS -1J -'--S.

Iu conclusion, we wish to thank Professor N. ]~. KUIPER and Dr. F. BRICKELL~ to both of whom we are indebted for some most helpful suggestions. R. S. CLARK - M. BRUCKI~IE/~IER: Tensor structures on a di]]erentiable, etc. 141

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