On hyper / structures

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We will be particularly interested in this last case, where the rank of E is three From now on, we will assume this cyclic permutation convention of indices for i,j,k, and we also set r,s,î = 1,2,3

Fixing an open set f/ С M, we define the local tensor fields

( 2 3) Ф1 =f!+Vj ^^k -m^^j,

and It is easy to check that they satisfy

Hence each фs determines an almost contact structure [Sa] Moreover, we have the relations

Ф1^ = ^Ь r],ФJ = ^Ь ФьФ} -r]j®ii = -Ф}Ф1 +Ц'^^]=Фк

Therefore iфsЛs^'ns)s=\ 2 3 determines а local almost contact 3-structure as defined by Kuo [Kuo] Then a hyper/-structure of corank m = 3 is precisely an almost contact 3-structure It is known that every almost contact 3-structure has an associated nemannian metric g that satisfies, for each s,

g { XJ ) = giфsX,ф,Y) + r]siX)r]s(Yl giès.X) = г]^(Х)

An easy calculation shows that this metric is also compatible with each /- structure, /, 1 e , It satisfies

g ( XJ ) = g(fsXJsY) + Y.'^r{X)Vr(Y)

Such a metric is not uniquely determined For a fixed compatible nemannian metric such structure is called a metric hyper/-structure From now on we will assume that we have chosen a compatible metric We now define the following 2-forms

uJs { X , Y ) = g(XJsY), Cüs(X,Y) = g(X,фsY),

then ojt =Lüi -r]j Ar]k, with our usual cyclic convention of indices We observe that the two forms uJs are linearly independent It is also clear that the two forms Lus are also hnearly independent

If Qs = drjs we have a contact 3-structure [Kuo] The structure will be called a 3-iS-structure if a;, = drjs

As was observed by Goldberg [Go], for a single globally framed/-manifold, *1 = ±(1/ |u;^ |)г71 Л 772 Л Л 77;n Л a;^ and о; is of rank 2k (for a manifold of dimension 2k+m) Since this relation holds for each uJs in our case, we conclude that

( 2 4) Г7] Л 772 Л 773 Л I ^ Л,а;, j ф О,

with (Al, Л2 Аз) G S^ each о;, of rank 4n In the case of a 3-5-structure,