Vol. 37 ( 2017 ) No. 3. J. of Math. (PRC) : A : (2017) k=1. ,, f. f + u = f φ, x 1. x n : ( ).

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1 Vol. 37 ( 2017 ) No. 3 J. of Math. (PRC) R N - R N - 1, 2 (1., ) (2., ) : R N., R N, R N -. : ; ; R N ; MR(2010) : 58K40 : O192 : A : (2017) [6], Mather f : (R n, 0) R p L, R, A, K... [2 5] J = L, R, C, K, [12]. ε n f : (R n, 0) R, m n f : (R n, 0) (R, 0), ε n. ε n m n, m n = m k n. f, j k f(x) f x k Taylor k=1., k =, j f(x) f x Taylor. f ε n, f R N - u m, φ f + u = f φ, f + m f R, R R n. [2] [5]. 1.1 f m n ε p n, : (1) f R - ; (2) J(f) m n ; (3) ( f ) f 0, J(f) Jacobian f x 1,, f x n ε n. : : : ( ). : (1977 ),,,, :.

2 468 Vol. 37, Sun Wilson [9]. [9], X x n -, I ε n n 1 x 1,, x n 1. f I 2, f + m I 2 f R X, f, R X R X,. 1.2 f I 2, f + m I 2 f R X m I J(f)., [7 8] R R N R I, R N R R X., R N, ; R R N, : f m n ε p n, f R N - m n ε p n T R N f, T R N f f. 2 ε n f : (R n, 0) R, m n f : (R n, 0) (R, 0), ε n. ε n m n, m n = m k n. f, j k f(x) f x k Taylor., k=1 k =, j f(x) f x Taylor. [6], Mather R = {h : (R n, 0) (R n, 0) }, f, g ε n R - h R, f h = g, h = (h 1, h 2,, h n ) T. h(0) = 0, h i (0) = 0, i = 1, 2,, n. h i (x) = n a ij (x)x j, x = (x 1, x 2,, x n ) T R n, a ij (x) ε n. h(x) = A[x] x, j=1 A(x) = (a ij (x)) n n. h x = 0 Jacobian J(h) x=0 = A[0], h, A[0]., f, g ε n R - A[x] ε n n n, A[0], f(a[x] x) = g(x). R = {A[x] A : (R n, 0) GL(R, n) }, R = {A[x] x A[x] R}. N R n n, R R ( [8]) N = I n, ( R N = {A[x] R A[x] N (A[x]) T = N, x (R n, 0)}, I r 0 0 I r, n = 2m N = R N = {A[x] x A[x] R N, x (R n, 0)}. ) ( ) ( 0 I m I m 0, R N, Lorentz R. ), R N = Sp(2m, R). ε p n f : (R n, 0) R p, m n ε p n f : (R n, 0) (R p, 0). f m n ε p n, V (f) f ε n. y 1 f,, y p f ε p n., f

3 No. 3 : R N - R N (R n, 0), (R p, 0), V (f) n ε n V (R n ), p ε p V (R p ). tf : V (R n ) V (f) tf(x)(x) = Df(x)(X(x)), X V (R n ), x (R n, 0), ε n, Df f Jacobian ( [10]). 2.1 ( [7]) f g R N - A[x] x R N f(a[x] x) = g(x) ( f A = g). 1 R N f f R N, f g R N - f g R N ( [7]) V N (R n ) = {A[x] A[x] ε n n n, A[x] N + N (A[x]) T = 0, x (R n, 0)}, V N (R n ) = {A[x] x A[x] V N (R n )}. 2.1 (Nakayama ) A ( 1), I A : α I, 1 + α A, M A -, N M A -. N + I M = M, N = M. 3 R N f f m n ε p n, R N f := {φ f φ R N } f R N. ε p n+1 γ, γ t (x) = γ(x, t), t, γ t ε p n. γ 0 = f γ t R N f, t, ψ t R N γ t = ψ t f, ψ t ψ : (x, t) ψ(x, t) = ψ t (x) (R n+1, 0) (R n, 0)., R N f, R k N jk f. 3.1 R k N = {jk φ φ R N }, V k N (Rn ) = {j k ϕ ϕ V N (R n )} (k 1), T e R k N e, T e R k N = V k N(R n ). R k Lie, RN k Rk, Lie Lie, RN k Lie. Rk N Rk N e T ern k, T e R k N = V k N(R n ) = V N (R n )/m k+1 n A[x] A[x] N + N (A[x]) T = 0. = {A[x] = (a ij (x)) n n a ij (x) J k (n, p)},, α : ( ε, ε) RN k, α(0) = I n, α(t) = A[t, x], dα(t) dt t=0 T e RN k, A[t, x] Rk N, A[t, x] N (A[t, x]) T = N. t, da[t,x] dt t=0 = A[0, x], A[0, x] V k N (Rn ). T e R k N V k N (Rn ). Ȧ[0, x] N + N ( A[0, x]) T = 0,

4 470 Vol. 37, A[x] V k N (Rn ), A[x] N + N (A[x]) T = 0. ε > 0, α : ( ε, ε) GL(R, n), α(t) = exp(ta), α(t) N α(t) T = exp(ta) N (exp(ta)) T = exp(ta) N (exp(ta T )) = exp(ta) (exp( ta)) N = exp((t t)a) N = N, A[x] N + N (A[x]) T = 0. α(t) RN k, T e RN k V N k(rn ). T e RN k = V N k(rn ). 2 T e R k N = V k N (Rn ). 3.1 ( [11]) G M, M. 3.2 f m n ε p n, k (k 1), R k N := {jk φ φ R N }, J k (n, p) := {j k f f ε p n}, R k N jk f j k f T jk fr k N j k f = tf(v N (R n ))/m k+1 n. R k N Lie, Rk N J k (n, p). j k f J k (n, p), R k N jk f j k f R k N, 3.1 Rk N jk f J k (n, p). R k N jk f j k f. R k N RN k T ern k = V N k(rn ) = {A[x] x A[x] VN k(rn )}. exp : T e RN k, T j fr k k N j k f = { d dt jk (f(exp(t(a[x] x)))) A[x] T e RN k = VN(R k n )}, t=0 A[x] x V k N (Rn ), 2, exp(t(a[x] x)) = g t R k N, d dt jk (f(exp(t(a[x] x)))) t=0 = j k ( d dt (f g t)(x) ) t=0 = j k (Df(x)(A[x] x)) tf(v N (R n ))/m k+1 n. 3.1 f m n ε p n, R N f f T R N f : T R N f = tf(v N (R n )). 3.2,. 4 R N - R N, R N -.

5 No. 3 : R N - R N f m n ε p n, f R N -, g m n ε p n j f(0) = j g(0), f g R N f m n ε p n, f R N - m n ε p n T R N f. ( ) f R N -. u m n ε p n. F (t, x) = f(x) + tu(x), t R F t (x) := F (t, x)., F t (x) R N f. F 0 (x) = f(x), df dt t=0 = u(x) T R N f. m n ε p n T R N f., [0, 1], g m n ε p n j f(0) = j g(0), F : (R R n, 0) R p F (t, x) = f(x) + t(g(x) f(x)), F t (x) := F (t, x). g R N - f. t, F t (x) R N - F t0 (x). F t0 (x) f(x) = (g(x) f(x)) g f m n ε p n+1, T R N F t0 T R N f, ( ), T R N f T R N F t0 + m n (T R N f), Nakayama,, T R N F t0 = T R N f. Ft = g(x) f(x) m n ε p n T R N f = T R N F t = tf t (V N (R n )),, ξ V N (R n ) ξ = n ξ i x i i=1 F t = tf t(ξ), F t(x) DF t (x)(ξ(x)) = 0, ( ) = (ξ 1,, ξ n ) = A[x] x, A[x] ε n n n, A[x] N +N (A[x]) T = 0, x (R n, 0), ( ) { dx 0 dt = 1, dx i dt = ξ i. Ãt(x) Ã ξ Ã t0 (x) = x, Ãt = A[Ã t (x)] Ã t (x). Ã 1 t (x) = A[x] x.

6 472 Vol. 37 A[x] ε n n n, h t (x). Ãt(x) = A[h t (x)] Ã t (x). ( [1]), Ã t (x) = e A[hs(x)]ds Ã t0 (x) = e A[hs(x)]ds x = lim µ 0 k=1 n e A[ht k (x)] tk x, µ [, t] t 1 t n, t k = t k t k 1. Ã t Ã t (x) = Ã t [x] x, Ã t [x] = t e A[hs(x)]ds. e A[ht k (x)] t k = n=0 e A[ht k (x)] tk N = N A[h t (x)] N + N (A[h t (x)]) T = 0, 1 ((A[h n! t k (x)]) t k ) n, 1 n! (( A[h t k (x)]) T t k ) n = N e (A[ht k (x)]) n=0 e A[hs(x)]ds N = N, Ã t [x] = t e A[hs(x)]ds, e (A[hs(x)]) T ds. T t k, Ã t [x] = A[h t (x)] Ã t [x] ( [1]) Ã t [x] (Ã 1 t [x]) T Ã 1 t [x] = A[h t (x)] Ã t [x] (Ã t [x]) T = (A[h t (x)]) T. 1 Ãt [x] = A[h t (x)] (Ã 1 t [x]) T = e (A[hs(x)]) T ds (Ã 1 [x]) T = Ã t [x] N = N (Ã 1 t [x]) T. Ã t [x] N (Ã t [x]) T = N. Ã t [x] R N. e (A[hs(x)]) T ds F t Ã t = F t Ã t + tf t Ã t = ( F t + tf t Ã t Ã 1 t ) Ã t = ( F t tf t(ξ)) Ã t = 0, d(ft Ãt) = 0. t, F t Ã t = F t0. [0, 1], [0, 1]., F t Ã t = F t0. [0, 1], F 1 Ã 1 = F 0, g Ã 1 = f, g R N - f.

7 No. 3 : R N - R N [1] John D D, Charles N F. Product integration with application to differential equation[m]. Boston: Addison Wesley Publ. Comp., [2] Wilson L C. Infinitely determined map-germs[j]. Can. J. Math., 1981, 33(3): [3] Wilson L C. Map-germs infinitely determined with respect to right-left equivalence[j]. Pacific J. Math., 1982, 102(1): [4] Broderesen H. A note of infinite determinacy of smooth map-germs[j]. Bull. London Math. Soc., 1981, 13: [5] Wall C T C. Finite determinacy of smooth map-germs[j]. Bull. London Math. Soc., 1981, 13: [6] Mather J N. Stability of C mappings iii: Finitely determined map-germ[j]. Pulb. Math. IHES., 1969, 35: [7]. A K [J]., 1999, 19(2): [8] Liu Hengxing. Finite indeterminacy of homogeneous polynomial germs under some subgroups R Ir of R[J]. Wuhan Univ. J. Nat. Sci., 2005, 10(5): [9] Sun B H, Wilson L C. Determinacy of smooth germs with real isolated line singularities[j]. Proceed. Amer. Math. Soc., 2001, 129(9): [10]. [M]. :, [11] Dimca A. Topics on real and complex singularities[m]. Braunschweig, Wiesbaden: Friedr. Vieweg Sohn, Adv. Lect. Math. Vieweg, [12] Zhang Guobin, Liu Hengxing. The infinite determinacy of function germs with parameters[j]. J. Math., 2005(3): THE INFINITE DETERMINACY AND TANGENT SPACE TO THE ORBIT OF SMOOTH MAP-GERMS UNDER THE ACTION OF R N SU Dan 1, LIU Heng-xing 2 (1.School of Statistics, University of International Business and Economics, Beijing , China) (2.School of Mathematics and Statistics, Wuhan University, Wuhan , China) Abstract: In this paper, the tangent space to the orbit of smooth map-germs under the action of R N is investigated. By produce integral theory, the infinite determinacy of smooth map-germs relative to a subgroup R N of right equivalent group is obtained, and the necessary and sufficient conditions for a smooth map-germ to be finitely R N -determined are extended. Keywords: map-germs; infinite determinacy; R N subgroup; tangent space 2010 MR Subject Classification: 58K40

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