We apply techniques of Painlevé–Kowalewski analysis to certain ODE reductions of the Ricci-flat equations. We particularly focus on two examples when the hypersurface is an Aloff–Wallach space or a circle bundle over a Fano product.

1.
Ablowitz
,
M.
,
Ramani
,
A.
, and
Segur
,
H.
, “
A connection between nonlinear evolution equations and ordinary differential equations of P-type I
,”
J. Math. Phys.
21
,
715
721
(
1980
);
Ablowitz
,
M.
,
Ramani
,
A.
, and
Segur
,
H.
, II.
J. Math. Phys.
21
,
1006
1015
(
1980
).
2.
Adler
,
M.
, and
van Moerbeke
,
P.
, “
Kowalewski’s asymptotic method, Kac-Moody Lie algebras and regularization
,”
Commun. Math. Phys.
83
,
83
106
(
1982
).
3.
Aloff
,
S.
, and
Wallach
,
N.
, “
An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
,”
Bull. Am. Math. Soc.
81
,
93
97
(
1975
).
4.
Bérard Bergery, L., “Sur de nouvelles variétés riemanniennes d’Einstein” (L’Institut Élie Cartan, Nancy, 1982).
5.
Boyer
,
C.
, and
Galicki
,
K.
, “
On Sasakian-Einstein geometry
,”
Int. J. Math.
11
,
873
909
(
2000
).
6.
Brandhuber
,
A.
,
Gomis
,
J.
,
Gubser
,
S.
, and
Gukov
,
S.
, “
Gauge theory at large N and new G2 holonomy metrics
,”
Nucl. Phys. B
611
,
179
204
(
2001
).
7.
Castellani
,
L.
,
D’Auria
,
R.
, and
Fré
,
P.
, “
SU(3)⊗SU(2)⊗U(1) from D=11 supergravity
,”
Nucl. Phys. B
239
,
610
652
(
1984
).
8.
Castellani
,
L.
, and
Romans
,
L. J.
, “
N=3 and N=1 supersymmetry in a new class of solutions for d=11 supergravity
,”
Nucl. Phys. B
238
,
683
701
(
1984
).
9.
Cleyton
,
R.
, and
Swann
,
A. F.
, “
Cohomogeneity-one G2-structures
,”
J. Geom. Phys.
44
,
202
220
(
2002
).
10.
Cvetic̆
,
M.
,
,
H.
,
Gibbons
,
G. W.
, and
Pope
,
C. N.
, “
New complete non-compact Spin(7) manifolds
,”
Nucl. Phys. B
620
,
29
54
(
2002a
).
11.
Cvetic̆
,
M.
,
,
H.
,
Gibbons
,
G. W.
, and
Pope
,
C. N.
, “
Cohomogeneity one manifolds of Spin(7) and G2 holonomy
,”
Ann. Phys.
300
,
139
184
(
2002b
).
12.
Dancer
,
A.
, and
Wang
,
M.
, “
Kähler-Einstein metrics of cohomogeneity one
,”
Math. Ann.
312
,
503
526
(
1998
).
13.
Dancer
,
A.
, and
Wang
,
M.
, “
The cohomogeneity one Einstein equations from the Hamiltonian viewpoint
,”
Reine Angew. Math.
524
,
97
128
(
2000
).
14.
Dancer
,
A.
, and
Wang
,
M.
, “
The cohomogeneity one Einstein equations and Painlevé analysis
,”
J. Geom. Phys.
38
,
183
206
(
2001
).
15.
Dancer, A. and Wang, M., “Painlevé expansions, cohomogeneity one metrics and exceptional holonomy,” preprint 2002.
16.
Dancer, A. and Wang, M., “Integrability and the Einstein equations,” in Symplectic and Contact Topology, edited by Y. Eliashberg, B. Khesin, and F. Lalonde (Fields Institute Communications, Toronto, 2003a).
17.
Dancer, A. and Wang, M., “Painlevé expansions and the Einstein equations: the two-summand case,” J. Geom. Phys. (to be published 2003b).
18.
Eschenburg
,
J.
, and
Wang
,
M.
, “
The initial value problem for cohomogeneity one Einstein metrics
,”
J. Geom. Anal.
10
,
109
137
(
2000
).
19.
Gray
,
A.
, “
Curvature identities for hermitian and almost hermitian manifolds
,”
Tohoku Math. J.
28
,
601
612
(
1976
).
20.
Gukov
,
S.
, and
Sparks
,
J.
, “
M-theory on Spin(7) manifolds
,”
Nucl. Phys. B
625
,
3
69
(
2002
).
21.
Kanno
,
H.
, and
Yasui
,
Y.
, “
On Spin(7) holonomy metric based on SU(3)/U(1) I, II
,”
J. Geom. Phys.
43
,
293
309
(
2002
);
Kanno
,
H.
, and
Yasui
,
Y.
,
J. Geom. Phys.
43
,
310
326
(
2002
).
22.
Page
,
D.
, and
Pope
,
C. N.
, “
New squashed solutions of d=11 supergravity
,”
Phys. Lett. B
147B
,
55
60
(
1984
).
23.
Smale
,
S.
, “
On the structure of 5-manifolds
,”
Ann. Math.
75
,
38
46
(
1962
).
24.
Wang
,
J.
, and
Wang
,
M.
, “
Einstein metrics on S2-bundles
,”
Math. Ann.
310
,
497
526
(
1998
).
25.
Wang
,
M.
, “
Some examples of homogeneous Einstein manifolds in dimension seven
,”
Duke Math. J.
49
,
23
28
(
1982
).
26.
Wang
,
M.
, and
Ziller
,
W.
, “
Einstein metrics on principal torus bundles
,”
J. Diff. Geom.
31
,
215
248
(
1990
).
27.
Witten
,
E.
, “
Search for a realistic Kaluza-Klein theory
,”
Nucl. Phys. B
186
,
412
428
(
1982
).
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