Differential geometry on almost tangent manifolds

  • 67 Accesses

  • 7 Citations


We start from a tensor field Q of type (1, 1) defined in a2n-dimensional manifold M which satisfies Q2=0 and has rank n. The tensor field Q defines an almost tangent structure in M. We then introduce another tensor field P of the same type and having properties similar to those of Q. We then define and study the tensors H=PQ, V=QP, J=P−Q, K=P+Q, L=PQ−QP, (J, K, L) defining an almost quaternion structure of the second kind on M. We study the differential geometry on almost tangent manifolds in terms of these tensors.


  1. [1]

    M. R. Bruckheimer,Thesis, University of Southampton, 1960.

  2. [2]

    E. Cartan,Les espaces de Finsler, Actualités Sci. et Ind.,79 (1934).

  3. [3]

    R. S. Clark -M. R. Bruckheimer,Sur les structures presque tangentes, C. R. Acad. Sci. Paris,251 (1960), pp. 627–629.

  4. [4]

    R. S. Clark -M. R. Bruckheimer,Tensor structures on a differentiable manifold, Annali di Mat. Pura ed Applicata, (IV),65 (1961), pp. 123–141.

  5. [5]

    R. S. Clark -D. S. Goel,Almost tangent manifolds of second order, Tôhoku Math. J.,24 (1972), pp. 79–92.

  6. [6]

    R. S. Clark - D. S. Goel,On the geometry of an almost tangent manifold, to appear in Tensor N. S.

  7. [7]

    E. T. Davies,On the curvature of the tangent bundle, Annali di Mat. Pura ed Applicata, (IV),81 (1969), pp. 193–204.

  8. [8]

    A. Deicke,Über die Darstellung von Finsler-Räumen durch nichtholonome Mannigfaltigkeiten in Riemannschen Räumen, Arch. Math.,4 (1953), pp. 234–238.

  9. [9]

    A. Deicke,Finsler spaces as non-holonomic subspaces of Riemannian spaces, J. of London Math. Soc.,30 (1955), pp. 53–58.

  10. [10]

    P. Dombrowski,On the geometry of the tangent bundle, J. regine und angew. Math.,210 (1962), pp. 73–88.

  11. [11]

    H. A. Eliopoulos,Structures presque tangentes dur les variétés différentiables, C. R. Acad. Sci. Paris,255 (1962), pp. 1563–1565.

  12. [12]

    H. A. Eliopoulos,On the general theory of differentiable manifolds with almost tangent structure, Canadian Math. Bull.,8 (1965), pp. 721–748.

  13. [13]

    A. Gray,Pseudo-Riemannian almost product manifolds and submersion, J. Math. and Mech.,16 (1967), pp. 715–737.

  14. [14]

    C. S. Houh,On a Riemannian manifold M 2n with an almost tangent structure, Canadian Math. Bull.,12 (1969), pp. 759–769.

  15. [15]

    A. Kandatu,Tangent bundle of a manifold with a nonlinear connection, Kōdai Math. Sem. Rep.,18 (1966), pp. 259–270.

  16. [16]

    O. Kowalski,Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. reine und angew. Math.,250 (1971), pp. 124–129.

  17. [17]

    A. J. Ledger -K. Yano,The tangent bundle of a locally symmetric space, J. London Math. Soc.,40 (1965), pp. 487–492.

  18. [18]

    J. Lehmann-Lejeune,Sur l'intégrabilité de certaines G-structures, C. R. Acad. Sci. Paris,258 (1964), pp.5326–5329.

  19. [19]

    P. Libermann,Sur les structures presque quaternioniennes de deuxième espèce, C. R. Acad. Sci. Paris,234 (1952), pp. 1030–1032.

  20. [20]

    A. Nijenhuis,X n−1-forming sets of eigenvectors, Proc. Kon. Ned. Acad. v. Wet., Series A,54 (1951), pp. 200–212.

  21. [21]

    B. O'Neill,The fundamental equations of a submersion, Michigan J. of Math.,13 (1966), pp. 459–469.

  22. [22]

    S. Sasaki,On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J.,10 (1958), pp. 338–354.

  23. [23]

    S. Tachibana -M. Okumura,On the almost complex structure of the tangent bundles of Riemannian spaces, Tôhoku Math. J.,14 (1962), pp. 156–161.

  24. [24]

    A. G. Walker,Connections for parallel distributions in the large, Quarterly J. of Math., (2),9 (1958), pp. 221–231.

  25. [25]

    K. Yano -M. Ako,Almost quaternion structures of the second kind and almost tangent structures, Kōdai Math. Sem. Rep.,25 (1973), pp. 63–94.

  26. [26]

    K. Yano -E. T. Davies,On the connection in Finsler space as an induced connection, Rend. Circ. Mat. Palermo, (2),3 (1955), pp. 409–417.

  27. [27]

    K. Yano -E. T. Davies,On the tangent bundles of Finsler and Riemannian manifolds, Rend. Circ. Mat. Palermo, (2),12 (1963), pp. 1–18.

  28. [28]

    K. Yano -S. Ishihara,Horizontal lifts of tensor fields and connections to the tangent bundle, J. Math. and Mech.,16 (1967), pp. 1015–1025.

  29. [29]

    K. Yano -S. Ishihara,Differential geometry of tangent bundles of order 2,Kōdai Math. Sem. Rep.,20 (1968), pp. 318–354.

  30. [30]

    K. Yano -S. Kobayashi,Prolongations of tensor fields and connections to the tangent bundle, I, II, J. of Math. Soc. Japan,18 (1966), pp. 194–210, 236–246.

  31. [31]

    K. Yano -A. J. Ledger,Linear connections on tangent bundles, J. of London Math. Soc.,39 (1964), pp. 495–500.

Download references

Additional information

To ProfessorBeniamino Segre on his seventieth birthday

Entrata in Redazione il 7 giugno 1973.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yano, K., Davies, E.T. Differential geometry on almost tangent manifolds. Annali di Matematica 103, 131 (1975) doi:10.1007/BF02414150

Download citation


  • Differential Geometry
  • Quaternion Structure
  • Tensor Field
  • Tangent Structure
  • Tangent Manifold