Submetric class の面積空間に於けるSchurの定理について
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(2) Vol. 18, No. 2 Journal of Hokkaido University of Education (Section BA) March, 1968. On the Theorem o? Schur in an Areal Space of the Submetric Class Masao GAMA Department of Mathematics, Asahikawa Branch,. Hokkaido University of Education. ^ H^ : Submetric class OW^SS^^M^^ Schur 05£ar^^-t. § 0. Introduction we consider the areal space which belongs to the submetric. class and is defined by (0. D (7,,, ^=0,1) IC^,-0 and 2^m^n-2,. where C^, ^ and K^jk are defined in C6] and C3] respectively. For 2^m^n—2 such the space may be said to be a more general space than a Minkowskian space defined. by 'ij> r\k = 0» ^ fvljh == 0.. . The purpose of this paper is to state that the theorem of Schur in Riemannian geometry CO holds good in our space. Throughout the paper we employ the. notations and results in the previous papers [2-] C3] C43 C53 without explanations. § 1. The curvature tensors In the previous paper C3] we exhibited the following properties of the curvature tensors in areal spaces of the submetric class : MjK== ~^i}ijk~l^i7l1cj. -lj? —•tv7tfcij —--o-^.v-'<j'> •Lv\hk —l-'7ifc» r ^'-, Ijf vr -£VA,/&A; ^~J7bJ&9 T \'ij. ^ T?r _l_ r" ^ J?r l_ /"' ^ Pr _L F A 7?r. 'ilcv r -l^-\hj "V '^'ihi •r •lv\jk ~V <-Jjfe) r •i^\l.h~T~'~Jjft> r JLY\kiJ). hjH, + ^ jfc/i, + ^ fr/tj — (-'ft; ? ^-\j1c ~ ^'J> r ^ \kh — ^K! r ^ \7rl == 0. which is one of the first forms of the Bianchi identities, Cl. 4) KhjlcM+KhKl\J+K'!}t,lJ\k+Phj, r K-'\U+Pl}i,^ ^^~Lj+JPL> ^K7\Jk=0, ~'1 \^A-T<r'1 r'1- ^—r<rl ft- ^_p( ^ _i_ 0' A !t,Jk | •/• ~T •lv hjl '-'1c, r —L^hkl'^j, r —t-h), •r\k^~ 1- M, r\J. 1) Throughout the paper Latin indices rum from 1 to n and Greek from 1 to m. 2) Numbers in brackets refer to the references at the end of the paper.. — 70 —.
(3) Masao Gama li jU, ps \ _ pi (A ps ^J_Q' ^ ^ ZTS ^ _. hj) s i~ (Afe i r ~ r Aifc ) s i~ 1^] > r ~T IJ h> s, r ^ iJ.jk =. which are two of the second forms of the Bianchi identities. In our space from (0. 1) it follows by virtue of the lemma in C53 and the relation ". _ C.IS-KT 3) -hjk =° L'rA^ sj-fc". ni jic=~ ^ jkht,. (1. 3)/ K^+K^+K^Q, (1. 4)/ K'^+K^+K^^Q, i/ T<T! \^-L.K~I rs ^—TT'- r'5 A-. hJH I r ~T tv- hjs^K, r — •"- hks'-'j, r ~ U.. § 2. The theorem of Schur As in Riemannian geometry we define the curvature. K by two vectors V\ and V[ as follows : (2. 1) K=-.. /'/i^T/jTT-fe 'hvif/r^vk • hj-'-lk ~a-Mi3ljJr \v lr 1r Z. In Riemannian geometry K represents the Gaussian curvature of the two-dimensional. geodesic subspace determined by the two vectors V\ and V[, In order that the curvature K be independent of both V\ and V[ it is necessary and sufficient that from (1. 1), (1. 2)/ and (1. 3)/ (2. 2) Z^,, = -XCV..- ^A,). Substituting the result obtained by the covariant differentiation of (2. 2) with respect to xl in (1. 4) and contracting with 9hj 9ik, we have. (2. 3) K^K, ,-K; ^=0. Similarly, substituting the result obtained by the covariant differentiation of (2. 2) with respect to p^ in (1. 5)/ and contracting with 9i1c9-i'-^, we have. (2. 4) K\^=K; ^=0. Finally, from (2. 3) and (2. 4) we conclude : Theorem The theorem of Schur in Riemannian geometry holds good also in the areal space ivhich belongs to the submetric class and is defined by (0. 1). REFERENCES 1) Eisenhart, L. P. (1926) Riemannian geometry. Princeton University Press. 2) Gama, ML (1965) On areal spaces of the submetric class. Tensor N. S. Vol. 16, No. 3, p. 262-268. 3) Gama, M. (1966) On areal spaces of the submetric class. III. Tensor N. S. Vol. 17, No. 1, p. 79-85. 4) Gama, M. (1967) On area! spaces of the submetric class. IV. Tensor N. S. Vol. 18, No. 1, p.. 3) See C41 — 71 —.
(4) On the Theorem of Schur in an Areal Space of the Submetric Class 49-53. 5) Gama, M. (1967) Some remarks on areal spaces of the submetric class, (in manuscript). 6) Kawaguchi, A. and Tandai, K. (1952) On areal spaces V. Normalized metric tensor and connection parameters in a space of the submetric class. Tensor N. S. Vol. 2, No. 1, p. 47-58.. — 72 —.
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