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Peter Michor
Peter Michor
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I think what you want are velocities (see 12.8 on page 120 of (1)). Iterated tangent bundles have lots of repetitions of the lower order parts. They and velocities are special cases of Weil bundles as treated in chapter VIII of (1). Elements of a Weil bundle act on functions as expansions (the generalization of derivation to the general case) at a point, see 37.6 of (1). On vector values functions they act coordinatewise.

The infinite dimensional version of Weil bundles is treated in section 31 of (2).

  • (1) Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) (pdf)

  • (2) Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

I think what you want are velocities (see 12.8 on page 120 of (1)). Iterated tangent bundles have lots of repetitions of the lower order parts. They and velocities are special cases of Weil bundles as treated in chapter VIII of (1). The infinite dimensional version of Weil bundles is treated in section 31 of (2).

  • (1) Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) (pdf)

  • (2) Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

I think what you want are velocities (see 12.8 on page 120 of (1)). Iterated tangent bundles have lots of repetitions of the lower order parts. They and velocities are special cases of Weil bundles as treated in chapter VIII of (1). Elements of a Weil bundle act on functions as expansions (the generalization of derivation to the general case) at a point, see 37.6 of (1). On vector values functions they act coordinatewise.

The infinite dimensional version of Weil bundles is treated in section 31 of (2).

  • (1) Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) (pdf)

  • (2) Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

Source Link
Peter Michor
Peter Michor
  • 25.3k
  • 2
  • 64
  • 112

I think what you want are velocities (see 12.8 on page 120 of (1)). Iterated tangent bundles have lots of repetitions of the lower order parts. They and velocities are special cases of Weil bundles as treated in chapter VIII of (1). The infinite dimensional version of Weil bundles is treated in section 31 of (2).

  • (1) Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) (pdf)

  • (2) Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)