Note on the locally product and almost locally product structures

  • Muppinaiya Nagaraj


This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold Xn of class Ck. Every locally product space has certain almost locally product structures which transform the local tangent space to Xn at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for Xn to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on Xn and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.


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  1. 1.
    Ficken, F. A... “The Riemannian and affine differential geometry of product spaces,”Annals of Mathematics, 1939,40, 892–913.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alfred Frolicher.. “Zur Differentialgeometrie der Komplexen und Fast-Komplexen Structuren,”Mathematische Annalen, 1955,129, 50–95.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Václav Hlavaty.. “An almost complex space of orderp and characteristic (n 1,n 2, …n p),”Tensor, 1960,10, 90–124.MATHMathSciNetGoogle Scholar
  4. 4.
    —————..Geometry of Einstein’s Unified Field Theory, P. Noordhoff, Ltd., Groningen, Holland, 1957.MATHGoogle Scholar
  5. 5.
    Schouten, J. A...Ricci Calculus, Second Edition, Springer Verlag, Berlin, 1954.MATHGoogle Scholar
  6. 6.
    Tracy Y. Thomas.. “Decomposition of Riemann spaces in the large,”Monatshefte für Mathematik und Physik, 1939, Band,49, 388–418.CrossRefGoogle Scholar
  7. 7.
    Kentaro Yano..Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, Macmillan and Co., New York, 1965.MATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 1967

Authors and Affiliations

  • Muppinaiya Nagaraj
    • 1
  1. 1.Department of Applied MathematicsIndian Institute of ScienceBangalore

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