This paper examines the initial data for the evolution of the space–time solution of Einstein’s equations admitting a conformal symmetry. Under certain conditions on the extrinsic curvature of the initial complete spacelike hypersurface and sectional curvature of the space–time with respect to sections containing the normal vector field, we have shown that the initial hypersurface is conformally diffeomorphic to a sphere or a flat space or a hyperbolic space or the product of an open real interval and a complete 2-manifold. It has been further shown that if the initial hypersurface is compact, then it is conformally diffeomorphic to a sphere. Finally, the conformal symmetries of a generalized Robertson–Walker space–time have been described.

1.
Alias
,
J.
,
Romero
,
A.
, and
Sanchez
M.
, “
Spacelike hypersurfaces of constant mean curvature in certain spacetimes
,”
Nonlinear Anal. Theory, Methods Appl.
30
,
655
(
1997
).
2.
Arnowitt
,
R.
,
Deser
,
S.
, and
Misner
,
C. W.
, “
The dynamics of general relativity
,”
Gravitation: An Introduction to Current Research
, edited by
L.
Witten
(
Wiley
, New York,
1962
)
3.
Beem
,
J. K.
and
Ehrlich
,
P. E.
,
Global Lorentzian Geometry
(
Marcel Dekker
, New York,
1981
).
4.
Berger
,
B. K.
, “
Homothetic and conformal motions in space-like slices of solutions of Einstein’s equations
,”
J. Math. Phys.
17
,
1268
(
1976
).
5.
Collinson
,
C. D.
and
French
,
D. C.
, “
Null tetrad approach to motions in empty spacetime
,”
J. Math. Phys.
8
,
701
(
1967
).
6.
Duggal
,
K. L.
and
Sharma
,
R.
,
Symmetries of Spacetimes and Riemannian Manifolds
(
Kluwer Academic
, Dordrecht,
1999
).
7.
Duggal
,
K. L.
and
Sharma
,
R.
, “
Conformal Killing vector fields on spacetime solutions of Einstein’s equations and initial data
,”
Nonlinear Anal. Theory, Methods Appl.
(to be published).
8.
Eardley
,
D.
,
Isenberg
,
J.
,
Marsden
,
J.
, and
Moncrief
,
V.
, “
Homothetic and conformal symmetries of solutions to Einstein’s equations
,”
Commun. Math. Phys.
106
,
137
(
1986
).
9.
Ferus
,
D.
, “
A remark on Codazzi tensors in constant curvature spaces
,”
Global Differential Geometry and Global Analysis
,
Lecture Notes in Math. 838
(
Springer-Verlag
, New York,
1981
).
10.
Garfinkle
,
D.
and
Tian
,
Q.
, “
Spacetimes with cosmological constant and a conformal Killing field have constant curvature
,”
Class. Quantum Grav.
4
,
137
(
1987
).
11.
Gerhardt
,
C.
, “
H-surfaces in Lorentzian manifolds
,”
Commun. Math. Phys.
89
,
523
(
1983
).
12.
Katsurada
,
Y.
, “
On a certain property of closed hypersurfaces in an Einstein space
,”
Comment. Math. Helv.
38
,
165
(
1964
).
13.
Kuehnel
,
W.
, “
Conformal transformations between Einsten spaces
,” in
Conformal Geometry
, edited by
R. S.
Kulkarni
and
U.
Pinkall
(
Vieweg Verlag Braunschweig
, Wiesbaden,
1988
).
14.
Maartens
,
R.
and
Maharaj
,
S. D.
, “
Conformal Killing vectors in Robertson-Walker spacetimes
,”
Class. Quantum Grav.
3
,
1005
(
1986
).
15.
Misner
,
C.
,
Thorne
,
K.
, and
Wheeler
,
J.
,
Gravitation
(
W. H. Freeman
, San Francisco,
1973
).
16.
O’Neill
,
B. O.
,
Semi-Riemannian Geometry with Applications to Relativity
(
Academic
, New York,
1983
).
17.
Sharma
,
R.
, “
Proper conformal symmetries of spacetimes with divergence-free conformal tensor
,”
J. Math. Phys.
34
,
3582
(
1993
).
18.
Yano
,
K.
,
Integral Formulas in Riemannian Geometry
(
Marcel Dekker
, New York,
1970
).
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