process

A Schwartz type algebra for the tangent groupoid 187with hW U ! R q a C 1 map such that h.x; 0/ D 0. Then1@FF.x;t/ D .x; 0/ C h.x; t/ t @from which we immediately get the result.Definition 3.3 (DNC). Let X M be as above. The set DX M 1provided with the Cstructure with border induced by the atlas described in the last proposition is called thedeformation to normal cone associated to X M . We will often write DNC insteadof deformation to the normal cone.Remark 3.4. Following the same steps, it is possible to define a deformation to thenormal cone associated to an injective immersion X,! M .Let us mention some basic examples of DCN manifolds D M X .Examples 3.5. 1. Consider the case when X D;. We have that D;M D M .0; 1with the usual C 1 structure on M .0; 1. We used this fact implicitly to cover DXMas in (4).2. Consider the case when X M is an open subset. Then we do not have anydeformation at zero and we immediately see from the definition that DXM is just theopen subset of M Œ0; 1 consisting of the union of X Œ0; 1 and M .0; 1.The most important feature of the DNC construction is that it is in some sensefunctorial. More explicitly, let .M; X/ and .M 0 ;X 0 / be C 1 -couples as above and letF W .M; X/ ! .M 0 ;X 0 / be a couple morphism, i.e., a C 1 map F W M ! M 0 withF.X/ X 0 . We define D.F /W DX M ! DM 0Xby the following formulas:0D.F /.x; ; 0/ D .F .x/; d N F x ./; 0/ and D.F /.m; t/ D .F .m/; t/ for t ¤ 0;where d N F x is by definition the map.N M X / xd N F x! .N M 0X 0/ F.x/induced by T x M dF x! T F.x/ M 0 .We have the following proposition, which is also an immediate consequence of thelemma above.Proposition 3.6. The map D.F /W DX M ! DM 0Xis C 1 .0Remark 3.7. If we consider the category C21 of C 1 pairs given by a C 1 manifold anda C 1 submanifold and pair morphisms as above, we can reformulate the propositionand say that we have a functorD W C 1 2 ! C 1where C 1 denotes the category of C 1 manifolds with border.