Publication: Homogeneous algebraic distributions
Loading...
Official URL
Full text at PDC
Publication Date
2001
Authors
Muñoz Masqué, Jaime
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Rocky Mountain Mathematics Consortium
Citations
Exportar
Abstract
Let p:E→M be a vector bundle of dimension n+m and (xλ,yi), λ=1,…,n, i=1,…,m, be fibre coordinates. A vertical vector field X on E is said to be algebraic [respectively, algebraic homogeneous of degree d] if its coordinate expression is of the type X=∑mi=1Pi∂/∂yi, where Pi are polynomials [respectively, homogeneous polynomials of degree d] in coordinates yi. A vertical distribution over E is said to be algebraic [respectively, homogeneous algebraic of degree d] if all local generators are homogeneous algebraic [respectively, homogeneous algebraic of the same degree d] vector fields. It is proved that a vertical distribution locally spanned by vector fields X1,…,Xr is homogeneous algebraic of degree d if and only if an r×r matrix A=(aij), aij∈C∞(E), exists which is equal to d−1 times the identity matrix along the zero section of E, and such that [χ,Xj]=∑ri=1aijXi, j=1,…,r, where χ is the Liouville vector field.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M.F Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
M.F Atiyah, Geometry and physics: Where are we going?, Lecture Notes in Pure. and Appl. Math. 184 (1997), 1–7.
D. Betounes, The geometry of gauge-particle field interaction: A generalization of Utiyama's theorem, J. Geom. Phys. 6 (1989), 107–125.
D. Bleecker, Gauge theory and variational principles, Addison-Wesley Publ. Co., Inc., Reading, MA, 1981.
N. Bourbaki, Éléments de mathématique, Algèbre, Hermann, Paris, 1970.
F. Cano, Réduction de singularités des feuilletages holomorphes, C.R. Acad. Sci. Paris 307 (1988), 795–798.
H. Cartan, Calcul différentiel, Hermann, Paris, 1967.
A. Grothendieck and J.A. Dieudonné, Éléments de géometrie algébrique, Springer-Verlag, Berlin, 1971.
V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge Univ. Press, Cambridge, UK, 1983.
I. Kolář, P.W. Michor and J. Slovák, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
P. Lecomte, On the infinitesimal automorphisms of a vector bundle, J. Math. Pures Appl. 60 (1981), 229–239.
P.A.B. Lecomte, P.W. Michor and H. Schikentanz, The multigraded Nijenhuis-Richardson algebra, its universal property and applications, J. Pure Appl. Algebra 77 (1992), 87–102. MR1148273 (93d:17036)
P. Libermann and Ch.M. Marle, Symplectic geometry and analytical mechanics, D. Reidel Publ. Co., Dordrecht, Holland, 1987. MR0882548 (88c:58016)
B. Malgrange, Frobenius avec singularitiés, Invent. Math. 39 (1977), 67–89. MR0508170 (58 #22685b)
Y.I. Manin, Gauge field theory and complex geometry, Springer-Verlag, Berlin, 1988. MR0954833 (89d:32001)
J. Martinet and J.P. Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. 55 (1981), 63–164. MR0672182 (84k:34011)
K. Michael, About the geometric structure of symmetry-breaking, J. Math. Phys. 32 (1991), 1065–1071.
J. Mickelsson, Current algebras and groups, Plenum Press, New York, 1989.
P.K. Mitter and C.M. Viallet, On the bundle of connections and the gaugeorbit manifold in Yang-Mills theory, Comm. Math. Phys. 79 (1981), 457–472.
J. Muñoz-Masqué and A. Valdés Morales, The number of functionally independent invariants of a pseudo-Riemannian metric, J. Phys. A: Math. Gen. 27 (1994), 7843–7855.
L. Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété, C.R. Acad. Sci. Paris 228 (1949), 1549–1551.
P. Olver, Equivalence, invariants and symmetry, Cambridge Univ. Press, Cambridge, UK, 1995. )
C. Roger, Algèbres de Lie graduées et quantification, in Symplectic geometry and mathematical physics, Prog. Math., Birkhauser, Boston, 1991, 374–421.
J.C. Tongeron, Ideaux de fonctions différentiables, Springer-Verlag, Berlin, 1972