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S-connection and Gauss-Codazzi equations

S-connection and Gauss-Codazzi equations Acta Mathematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 75~80. By M. D. UPADHYAY and C. P. AWASTHI (Lucknow) 1. Preliminaries. Let us consider a Riemannian space Is, of coordinates x i (i= 1, ..., n) ~ immersed in a Riemannian space V, + ~ of coordinates y~ (c~ = 1, ..., n + 1)? Let the metrics of V,, and V,+~ be positive definite and given by g~jdx~d# and a~Bdy~dy ~, respectively. Then we have ,a fl (1.1) gij = a~py;iy;j. Kz-curves were given by TSAGAS [1] as curves for which the unit vector field ~ along the congruence of curves, belong to the plane formed by the geodesic curvature vectors of K~-curve with respect to V,+a and V,. We have [1], (1.2) pk _ g i ~ t~ pJ (1 _ r~) - ~ t ~ = O. If s is the arc length of the curve, the intrinsic derivative of the unit tangent dx i . ds m the direction of the curve has components [2, p. 160], (1.3) p~= d~x~ {i} dxj dxk --J2 § jk ds ds " 2. S-connection. From (1.2) and (1.3) we have ~- --g~jt~PJ(1--r~)-ltkgtm ~ -~s -- O. or (2.1) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

S-connection and Gauss-Codazzi equations

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895949
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 75~80. By M. D. UPADHYAY and C. P. AWASTHI (Lucknow) 1. Preliminaries. Let us consider a Riemannian space Is, of coordinates x i (i= 1, ..., n) ~ immersed in a Riemannian space V, + ~ of coordinates y~ (c~ = 1, ..., n + 1)? Let the metrics of V,, and V,+~ be positive definite and given by g~jdx~d# and a~Bdy~dy ~, respectively. Then we have ,a fl (1.1) gij = a~py;iy;j. Kz-curves were given by TSAGAS [1] as curves for which the unit vector field ~ along the congruence of curves, belong to the plane formed by the geodesic curvature vectors of K~-curve with respect to V,+a and V,. We have [1], (1.2) pk _ g i ~ t~ pJ (1 _ r~) - ~ t ~ = O. If s is the arc length of the curve, the intrinsic derivative of the unit tangent dx i . ds m the direction of the curve has components [2, p. 160], (1.3) p~= d~x~ {i} dxj dxk --J2 § jk ds ds " 2. S-connection. From (1.2) and (1.3) we have ~- --g~jt~PJ(1--r~)-ltkgtm ~ -~s -- O. or (2.1)

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

References

  • K λ-curves of a hypersurface of a Riemannian space
    Tsagas, G.