Continuity of projections of natural bundles

• Volume: 57, Issue: 2, page 105-120
• ISSN: 0066-2216

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Abstract

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This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M, a topological space N, a group homomorphism E: Diff(M) → Homeo(N) and a function π: N → M is called a quasi-natural bundle if (1) π ∘ E(f) = f ∘ π for every f ∈ Diff(M) and (2) if f,g ∈ Diff(M) are two diffeomorphisms such that f|U = g|U for some open subset U of M, then E(f)|π^{-1}(U) = E(g)|π^{-1}(U). We give conditions which ensure that π: N → M is continuous. In particular, if (M,N,E,π) is a quasi-natural bundle with N Hausdorff, then π is continuous. Using this result, we classify (quasi) prolongation functors with compact fibres.

How to cite

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Włodzimierz M. Mikulski. "Continuity of projections of natural bundles." Annales Polonici Mathematici 57.2 (1992): 105-120. <http://eudml.org/doc/262285>.

@article{WłodzimierzM1992,
abstract = {This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M, a topological space N, a group homomorphism E: Diff(M) → Homeo(N) and a function π: N → M is called a quasi-natural bundle if (1) π ∘ E(f) = f ∘ π for every f ∈ Diff(M) and (2) if f,g ∈ Diff(M) are two diffeomorphisms such that f|U = g|U for some open subset U of M, then E(f)|π^\{-1\}(U) = E(g)|π^\{-1\}(U). We give conditions which ensure that π: N → M is continuous. In particular, if (M,N,E,π) is a quasi-natural bundle with N Hausdorff, then π is continuous. Using this result, we classify (quasi) prolongation functors with compact fibres.},
author = {Włodzimierz M. Mikulski},
journal = {Annales Polonici Mathematici},
keywords = {natural bundle; quasi-natural bundle; regular quasi-natural bundle; locally determined associated space; quasi-prolongation functor; manifold; prolongation functors},
language = {eng},
number = {2},
pages = {105-120},
title = {Continuity of projections of natural bundles},
url = {http://eudml.org/doc/262285},
volume = {57},
year = {1992},
}

TY - JOUR
AU - Włodzimierz M. Mikulski
TI - Continuity of projections of natural bundles
JO - Annales Polonici Mathematici
PY - 1992
VL - 57
IS - 2
SP - 105
EP - 120
AB - This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M, a topological space N, a group homomorphism E: Diff(M) → Homeo(N) and a function π: N → M is called a quasi-natural bundle if (1) π ∘ E(f) = f ∘ π for every f ∈ Diff(M) and (2) if f,g ∈ Diff(M) are two diffeomorphisms such that f|U = g|U for some open subset U of M, then E(f)|π^{-1}(U) = E(g)|π^{-1}(U). We give conditions which ensure that π: N → M is continuous. In particular, if (M,N,E,π) is a quasi-natural bundle with N Hausdorff, then π is continuous. Using this result, we classify (quasi) prolongation functors with compact fibres.
LA - eng
KW - natural bundle; quasi-natural bundle; regular quasi-natural bundle; locally determined associated space; quasi-prolongation functor; manifold; prolongation functors
UR - http://eudml.org/doc/262285
ER -

References

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8. [8] W. M. Mikulski, Locally determined associated spaces, J. London Math. Soc. (2) 32 (1985), 357-364.
9. [9] D. Montgomery and L. Zippin, Transformation Groups, Interscience, New York 1955. Zbl0068.01904
10. [10] A. Nijenhuis, Natural bundles and their general properties, in: Diff. Geom. in honor of K. Yano, Kinokuniya, Tokyo 1972, 317-334.
11. [11] R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology 16 (1978), 271-277. Zbl0359.58004
12. [12] S. E. Salvioli, On the theory of geometric objects, J. Differential Geom. 7(1972), 257-278. Zbl0276.53013
13. [13] J. Slovák, Smooth structures on fibre jet spaces, Czechoslovak Math. J. 36 (111) (1986), 358-375. Zbl0629.58001

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