The following three geometrical structures on a manifold are studied in detail: Leibnizian: a nonvanishing one-form Ω plus a Riemannian metric 〈⋅,⋅〉 on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. Galilean: Leibnizian structure endowed with an affine connection ∇ (gauge field) which parallelizes Ω and 〈⋅,⋅〉. For any fixed vector field of observers Z(Ω(Z)≡1), an explicit Koszul-type formula which reconstructs bijectively all the possible ∇’s from the gravitational G≔∇ZZ and vorticity ω≔12 rot Z fields (plus eventually the torsion) is provided. Newtonian: Galilean structure with 〈⋅,⋅〉 flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and ω≡0). Classical concepts in Newtonian theory are revisited and discussed.

1.
Bernal
,
A. N.
,
López
,
M.
, and
Sanchez
,
M.
, “
Fundamental units of length and time
,”
Found. Phys.
32
,
77
108
(
2002
).
2.
Cartan
,
E. E.
, “
Sur les variétés à connexion affine et la théorie de la relativité generalisée
Ann. Scient. Ecole Norm. Sup.
40
,
325
412
(
1923
).
3.
Dixon
,
W. G.
, “
On the uniqueness of the Newtonian theory as a geometric theory of gravitation
,”
Commun. Math. Phys.
45
,
167
182
(
1975
).
4.
Duval
,
C.
,
Burdet
,
G.
,
Künzle
,
H. P.
, and
Perrin
,
M.
, “
Bargmann structures and Newton-Cartan theory
,”
Phys. Rev. D
31
,
1841
1853
(
1985
).
5.
Ehlers, J., “The nature and structure of spacetime,” in The Physicist’s Conception of Nature, edited by J. Mehra (Reidel, Dordrecht-Holland, 1973), pp. 71–91.
6.
Ehlers
,
J.
and
Buchert
,
T.
, “
Newtonian cosmology in Lagrangian formulation: foundations and perturbation theory
,”
Gen. Relativ. Gravit.
29
,
733
764
(
1997
).
7.
Gautreau
,
R.
, “
General relativity in Newtonian form
,”
Gen. Relativ. Gravit.
22
,
671
681
(
1990
).
8.
Keskinen
,
R.
and
Lehtinen
,
M.
, “
On the linear connection and curvature in Newtonian mechanics
,”
J. Math. Phys.
17
,
2082
2084
(
1976
).
9.
Kobayashi, S., Transformation Groups in Differential Geometry (Springer-Verlag, New York, 1972).
10.
Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry (Interscience–Wiley, New York, 1963).
11.
Kunzle
,
H. P.
, “
Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics
,”
Ann. Inst. Henri Poincare, Sect. A
17
,
337
362
(
1972
).
12.
Kunzle
,
H. P.
, “
Covariant Newtonian limit of Lorentz space-times
,”
Gen. Relativ. Gravit.
7
,
445
457
(
1976
).
13.
Misner, C. W., Thorne, K. S., and Wheeler, J. A. Gravitation (Freeman, San Francisco, 1970).
14.
Navarro
,
J. A.
and
Sancho de Salas
,
J. B.
, “
The structure of the Newtonian limit
,”
J. Geom. Phys.
44
,
595
622
(
2003
).
15.
Rendall
,
A.
, “
The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system
,”
Commun. Math. Phys.
163
,
89
112
(
1994
).
16.
Rodrigues
,
W. A.
,
Souza
,
Q.
, and
Bozhkov
,
Y.
, “
The mathematical structure of Newtonian spacetime: classical dynamics and gravitation
,”
Found. Phys.
25
,
871
924
(
1995
).
17.
Sachs, R. K. and Wu, H. H., General Relativity for Mathematicians (Springer-Verlag, New York, 1977).
18.
Sonego
,
S.
and
Massar
,
M.
, “
Covariant definition of inertial forces: Newtonian limit and time-dependent gravitational fields
,”
Class. Quantum Grav.
13
,
139
144
(
1996
).
19.
Spivak, M., A Comprehensive Introduction to Differential Geometry (Publish or Perish, Waltham, MA 1979).
20.
Strichartz
,
R. S.
, “
Sub-Riemannian geometry
,”
J. Diff. Geom.
24
,
221
263
(
1986
).
21.
Trautman, A., “Comparison of newtonian and relativistic theories of space-time” in Perspectives in Geometry and Relativity, edited by B. Hoffmann (Indiana University Press, Bloomington, 1966).
22.
Wald, R. M., General Relativity, (University of Chicago, Chicago, 1984).
23.
Warner, F., Foundations of Diferential Manifolds and Lie Groups (Scott-Foresman, Glenview, IL, 1971).
This content is only available via PDF.
You do not currently have access to this content.