Basic Mathematical Tools

  • Maciej Błaszak


In this chapter we briefly discuss some elements of differential calculus which are important for understanding the content of this book. The reader who is familiar with the theory of tensor fields, Riemannian geometry and symplectic (Poisson) geometry can skip that part, keeping in mind that all important formulas of these formalisms are collected in this chapter. The reader who is less familiar with these mathematical tools will find here necessary knowledge presented in a compact form. For a more comprehensive treatment of the subject we refer the reader to the literature.


  1. 1.
    Abraham, R., Marsden, J.E.: Fundation of Mechanics, 2nd edn. Benjamin/Cummings, New York (1978)Google Scholar
  2. 54.
    Bordemann, M., Neumaier, N., Waldmann, S.: Homogeneous fedosov star products on cotangent bundles I: weyl and standard ordering with differential operator representation. Commun. Math. Phys. 198, 363 (1998)MathSciNetCrossRefADSGoogle Scholar
  3. 55.
    Bordemann, M., Neumaier, N., Waldmann, S.: Homogeneous Fedosov star products on cotangent bundles II: GNS representations, the WKB expansion, traces, and applications. J. Geom. Phys. 29, 199 (1999)MathSciNetCrossRefADSGoogle Scholar
  4. 71.
    Crampin, M., Pirani, F.A.E.: Applicable Differential Geometry. Lecture Note Series 59. Cambridge University, Cambridge (1988)Google Scholar
  5. 116.
    Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University, Cambridge (2006)CrossRefGoogle Scholar
  6. 119.
    Fedosov, B.V.: Deformationquantization and Index Theory. Akademie, Berlin (1996)zbMATHGoogle Scholar
  7. 178.
    Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. D. Reidel, Dordreht (1987)CrossRefGoogle Scholar
  8. 202.
    Mok, K.-P.: Metric and connections on the cotangent bundle. Kodai Math. Semin. Rep. 28, 226 (1977)MathSciNetCrossRefGoogle Scholar
  9. 219.
    Plebański, J.F., Przanowski, M., Turrubiates, F.J.: Induced symplectic connection on the phase space. Acta Phys. Polon. B 32, 3 (2001)MathSciNetzbMATHADSGoogle Scholar
  10. 230.
    Schouten, J.A.: Über Differentialkomitanten zweier kontravarianten Grössen. Proc. Akad. Wet. Amsterdam 43, 449 (1940)zbMATHGoogle Scholar
  11. 251.
    Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. In: Progress in Mathematics. Birkhäuser, Basel (1994)Google Scholar
  12. 255.
    von Westenholz, C.: Differential forms in mathematical physics. In: Studies in Mathematics and its Applications, vol. 3. North-Holland Publishing Company, North-Holland (1981)Google Scholar
  13. 258.
    Wasserman, R.H.: Tensor and Manifolds with Applications to Physics, 2nd edn. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  14. 267.
    Yano, K., Ishihara, S.: Tangent and cotangent bundles. Marcel Dekker, Inc., New York (1973)zbMATHGoogle Scholar
  15. 268.
    Yano, K., Patterson, E.M.: Vertical and complete lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan 19, 91 (1967)MathSciNetCrossRefGoogle Scholar
  16. 271.
    Zarraga, J.A., Perelomov, A.M., Perez Bueno, J.C.: The Schouten Nijenhuis bracket, cohomology and generalized Poisson structures. J. Phys. A Math. Gen. 29, 7993 (1996)MathSciNetCrossRefADSGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maciej Błaszak
    • 1
  1. 1.Division of Mathematical PhysicsFaculty of Physics UAMPoznańPoland

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