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 equi- valent 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 <t diffcr¢';ll~tbb'. ~qc. If M admits two tensor structures, then it is possible that two represen- tative 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 qua- ternionic 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 connec- tion 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 me- tric is said to be related to a given tensor structure if the two structures have a common subordinate structure. "We have defined a normal representa- tive 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 inde- pendent 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 defi- nes 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 subor- dinate 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 diffe- rentiable. 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 sub- group 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-sin- gular 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 neighbour- hood 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).