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On a class of Hilbert C*-manifolds 2213 The invariant connection3.1. The definition of a connection for Banach manifolds cannot be, due to the scarcityof smooth functions, the usual one. We follow [6], 1.5.1 Definition.Definition 3.2. Let M be a manifold, modeled over the Banach space E, and denotethe space of bounded bilinear mappings E E ! E by L 2 .E; E/. Then M is saidto possess a connection iff there is an atlas U for M so that for each U 2 U there is asmooth mapping W U ! L 2 .E; E/, called the Christoffel symbol of the connectionon U , which under a change of coordinates ˆ transforms according to.ˆ0X;ˆ0Y/D ˆ00 .X; Y / C ˆ0.X;Y/:The covariant derivative of a vector field Y in the direction of the vector field X is,locally, defined to be the principal part ofr X Y D dX.Y /.X;Y/;where, in a chart, the principal part of .u; X/ 2 U T.U/is X.The reader should note that this definition is equivalent to specifying a smoothvector subbundle H of TTM with the property that H p is for each p 2 TM closedand complementary to the tangent space ker.d p / of the fiber E .p/ through p. Itisnot, however, equivalent to the requirement that r be C 1 .M /-linear in its first variableand a derivation w.r.t. the action of C 1 .M / on vector fields, although connections asdefined above do have this property.3.3. We can keep the notion of invariance of a connection under the smooth action ofa (Banach) Lie group, however. In fact, if such a group G acts on M then, for eachg 2 G a connection g r is defined by lettingg r X Y Dr g Xg Y;g X.gm/ D d m gX.m/:Christoffel symbols then transform as in the definition above, g.m/ .g 0 .m/X.m/; g 0 .m/Y.m// D g 00 .m/.X.m/; Y.m// C g 0 .m/ m .X.m/; Y.m//;and we call r invariant under the action of G whenever g rDrfor all g 2 G.3.4. If M is the Hilbert C*-manifold U that was defined in the last section, therecan be at most one connection which is invariant under the action of the group ofbiholomorphic self-maps of U . This is because the difference of two of them is thedifference of their Christoffel symbols which have to vanish at each point according tothe way they transform under the reflections a D M a o M a , o .x/ D x. To findone, we let o .x; y/ D 0:Then, for any g leaving the origin fixed, g 00 D 0 and so 0 remains zero when transformedunder g. Using the transformation rule for Christoffel symbols we find