Submetric class の面積空間についての注意

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(1)Title. Submetric class の面積空間についての注意. Author(s). 蒲, 雅夫. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 18(1) : 1-3. Issue Date. 1967-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5877. Rights. Hokkaido University of Education.

(2) Vol. 18, No. 1 Journal of Hokkaido University of Education (Section II A) September 1967. Some Remarks on Areal Spaces of the Submetric Class. Masao GAMA Department of MEathematics, Asahigawa Branch, Hokkaido University of Education. Y'l ^^ : Submetric class (D M^ÎW ^01 ^ -C 0 '^^. 0. Introduction. We consider an areal space Are2) of the submetric class the metric tensor of which is the normalized metric tensor. In A^) we can express the metric bitensor gij,jci given by A. Kawaguchi [6]° in terms of the normalized metric tensor. The principal purpose of the paper is to give the conditions for the covariant derivative of the metric bitensor gij,u expressed in terms of the normalized metric tensor with respect to xft to vanish. We employ the same notations as in the previous papers [1] [2] [3]. Throughout the paper, Latin indices run from I to n and Greek indices run from I to 2. I. Condition for degeneration of the metric bitensor. The metric bitensor gij,ki is expressed making use of the Legendre's form as follows :. (1.1) g^4F^L^L^+4L^PW]+p[iP[lP]]P^ In our space, since the Legendre's form is expressed as a>|3 _ <T" o-niP i o-a!S^~iP_2). /lj = Slj S "' -h & ' L-OS" ' ,. (1,1) is reformed as follows :. (1.2) gw=2g^gm+AWip[ig^+Cmk,~)C^. From (1.2) we conclude :. In order that the metric bitensor gijî degenerate to 2g^kgj^ it is necessary and sufficient that the tensor C^ vanish. This shows that Theorem by H. Iwamoto [5] is valid for degeneration of the metric bitensor gij,w.. 2. Conditions for vanishing of the covariant derivative of the metric bitensor. Let us differentiate covariantly (1.2) with respect to xft. Then we have gij,M\Ji, == 8 ^2p^p[i ^[]|a)| +C[i^[i) C'j]^2]|7t from which we conclude Theorem. In order that the covariant derivative of the metric bitensor gij,w, with respect. to xh vanish, it is necessary and sufficient that the covariant derivative of the tensor C^ with respect to x11 vanish. 1) Numbers in brackets refer to the references at the end of the paper.. 2) See [7].. ( 1).

(3) Some Remarks on Areal Spaces of the Submetric Class. This result coincides essentially with the one by T. Igarashi [4].. Corollary 1. In order that the covariant derivative of the metric bitensor gij^ with respect to xh vanish, it is necessary and sufficient that the covariant derivative of the Legendre's form with respect to xh vanish. Proof. Corollary 1 can be proved in virtue of Theorem since al(3 __ âis/^'-p.. •i'ffi^g ^l'JS\h-. Corollary 2. If A,a is an ciffinely-connected space, the covariant derivative of the metric bitensor gijî tuith respect to xh vanishes. Proof. From the following relations. (2.2) C,^ = -|-C,.4(^ ^+r? /3j- r? ^) and. (2,3) C^ = Crs^W Ssj- r\ ^ +r, ^ ^. which is derived from (2.2), we see that Cij^h =0 is equivalent to Cy,^|/i,=0. Consequently in virtue of Theorem by S. Kikuchi [8] and (2.1), we have Corollary 2. This can also be proved in virtue of Theorem in [2].. Remark. From (2.2} and {2.3} it is concluded that Are ;'s a Riemannian space if and only if the tensor Cij^ vanishes. Concerning the curvature tensor PL-,^ defined in [3], we have. Lemma. In order that Aw (2^m^%-2) be an a ffinely-connected space, it is necessary and sufficient that the curvattire tensor Pmcaj vanish. Proof. Since the necessity is evident in virtue of Theorem by S. Kikuchi [8J and the suffi.cien.cy can be proved in the same way as Theorem 3.1 by K. Tandai [9]. Corollary 3. In An2) if the curvature tensor PL,^ vanishes and n ^ 4, ^/ze covariant derivative of the metric bitensor gij,M zvith respect to xh vanishes. Proof. Corollary 3 is an immediate consequence of Corollary 2 and Lemma.. Corollary 4. // A^2) ?s a JVLinkowskian space, the covariant derivative of the metric bitensor gij,^ with respect to x'1 vanishes. Proof. If A^ is a IVIinkowsldan space, Ci]c?j\ii=0 and, therefore, from (2.1) and (2.3) Ci'JîÔ which proves Corollary 4.. REFERENCES 1. Gama, M. (1965), On areal spaces of the submetric class. Tensor, N. S. Vol. 16, p. 262-268. 2. Gama, M. (1965), On areal spaces of the submetric class II. Tensor, N. S. Vol. 16, p. 291-293. 3. Gama, M. (1966), On areal spaces of the submetric class III. Tensor, N. S. Vol. 17, p. 79-85. 4. Igarashi, T. (1966), Some remarks on the af&nely connected areal space. Memoirs of the Muroran institute of Technology, Vol. 5, p. 419-422. 5. Iwamoto, H. (1948), On geometries associated with multiple integrals. Math. Japonicae, Vol. 1, p. 74-91. 6. Kawaguchi, A. (1950), On areal spaces I, Metric tensors in n-dimensional spaces based on the notion of two-dimensional area. Tensor, N. S. Vol. 1, p. 14-45.. (2).

(4) Masao Gama 7. Kawaguchi, A. and Tandai, K. (1952), Normalized metric tensor and connection parameters in a space of the submetric class. Tensor, N. S. Vol. 2, p, 47-58. 8. Kikuchi, S. (1966), Some remarks on connections in an areal space of the submetric class. Tensor, N. S, Vol. 17, p. 44-48. 9. Tandai, K. (1954), On areal spaces VII, The theory of the canonical connection and w-dimensional subspaces. Tensor, N. S. Vol. 4, p. 78-90.. C3).

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