Zoran Skoda
formal geometry

Formal geometry and Borel’s theorem



Let MM, NN be real smooth manifolds, U,VU,V open subsets of NN, aUVMa\in U\cap V\subset M and f:UNf\colon U\to N, g:VNg\colon V\to N smooth maps. We say that ff is tangent of order kk\in\mathbb{N} to gg and we right f kgf\sim_k g if f(a)=g(a)f(a) = g(a) and for some (therefore every) pair of smooth charts ϕ:W m\phi\colon W\to\mathbb{R}^m around aa, such that ϕ(a)=0\phi(a)=0, ψ:W n\psi\colon W'\to\mathbb{R}^n, ψ(f(a))=0\psi(f(a)) = 0, we have

ψ(f(x))ψ(g(x))h(x)ϕ(x)ϕ(a) k,xWf 1(W)g 1(W), \|\psi(f(x)) - \psi(g(x))\| \leq h(x)\|\phi(x)-\phi(a)\|^k, \,\,\,\,\,\,\forall x\in W\cap f^{-1}(W')\cap g^{-1}(W'),

where h(x)h(x) is a function defined and continuous on Wf 1(W)g 1(W)W\cap f^{-1}(W')\cap g^{-1}(W') and such that h(x)0h(x)\to 0 whenever xax\to a. Equivalently, the Taylor series of all componenets of ff and gg agree up to order kk. Tangency of order kk at point xx is a relation of equivalence on the set of smooth functions locally defined around xx. The corresponding class of equivalence ff is the kk-jet_ j k(f)j^k(f). kk-jets of smooth maps MNM\to N at point xMx\in M form a set of kk-jets J k(x)=J k(M,N,x)J^k(x)=J^k(M,N,x), with canonical structure of a real vector space. Note the obvious projections J k+1(x)J k(x)J^{k+1}(x)\to J^k(x); these define an inverse system of vector spaces. Its limit is the space J (x)=J (M,N,x):=lim kJ k(M,N,x)J^\infty(x) = J^\infty(M,N,x)\colon = \lim_k J^k(M,N,x) of \infty-jets at xx of smooth maps from MM to NN. A formal function is a an \infty-jet at xx of smooth functions from MM to \mathbb{R}.

(Emile Borel’s theorem on formal power series.) Let 𝒞 x \mathcal{C}^\infty_x be the vector space of germs of smooth real functions around a point x nx\in\mathbb{R}^n. Then a linear map can:𝒞 x J ( n,,x)\mathrm{can}\colon\mathcal{C}^\infty_x\to J^\infty(\mathbb{R}^n,\mathbb{R},x) of taking the Taylor series at xx is a surjection.

Taylor series of a smooth function does not need to converge and if it does it may converge to a different function. For example, for the Cauchy function (equal e 1/x 2e^{-1/x^2} for x0x\neq 0 and equal 11 for x=0x=0) all Taylor coefficients at 00 vanish. Of course, the germs of real analytic functions around xx could be identified with the corresponding formal functions: the restriction of can\mathrm{can} to the subspace of all germs of analytic functions is injective.

Differential operators

Let RR be a commutative unital kk-algebra. For every rRr\in R denote by l(r)l(r) the kk-linear map l(r)(r)=rrl(r)(r') = r\cdot r' of muliplication by rr from the left. The function rl(r)r\mapsto l(r) defines a morphism of rings l:REnd k(R)l\colon R\to\operatorname{End}_k(R). For fEnd k(R)f\in End_k(R) let us define [f,r]:=fl(r)l(r)fEnd k(R)[f,r] := f\circ l(r) - l(r)\circ f\in\operatorname{End}_k(R). We say that (not necessarily cocomplete!) filtration F 0F 1F 2End kRF_0\subset F_1\subset F_2\subset \ldots \subset End_k R is diferencial filtration if [f,r]F k[f,r] \in F_k for each fF k+1f\in F_{k+1} and for all k 0k\in\mathbb{N}_0. Ring End k(R)End_k(R) has a maximal differential filtration in the sense of inclusion at each filtered level; its colimit is by definition the subring of regular differential operators Diff(R)End k(R)Diff(R)\subset End_k(R).

Regular diferential operator of order s 0s\in\mathbb{N}_0 is a kk-linear map DEnd k(R)D\in\operatorname{End}_k(R) such that for all r 0,r 1,,r sRr_0, r_1,\dots,r_s\in R we have [[D,r 0],r 1],,r s]=0[\ldots[D,r_0],r_1],\ldots,r_s]=0. Regular diferential operators of the order ss form a subspace Diff s(R)End kRDiff_s(R)\subset End_k R; regular diferential operator is an element of the space Diff(R)= s=0 Diff s(R)Diff(R) = \cup_{s=0}^\infty Diff_s(R). The left multiplication l(r)l(r) by the element rRr\in R is a regular diferencial operator of order 00; thus, if RR is an integral domain, then RDiff 0(R)R\subset\operatorname{Diff}_0(R). Diferential filtration Diff 0(R)Diff 1Diff k(R)End k(R)Diff_0(R)\subset Diff_1 \subset\ldots\subset Diff_k(R) \subset\ldots\subset End_k(R) is typically not cocomplete, i.e. its union
Diff(R)End k(R)Diff(R)\neq End_k(R). C C^\infty-diferential operator on C C^\infty-manifold MM is a regular diferencial operator over C (M)C^\infty(M). They have the tensoriality property, i.e. if f| U=0f|_U=0 on open subspace UU, then (Df)| U=0(D f)|_U = 0.


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  • S. Helgason, Differential geometry, Lie groups and symmetric spaces, Acad. Press 1978; Amer. Math. Soc. 2001.

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