Abstract
There is a rich theory of so-called (strict) nearly Kähler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the $6$-sphere induced by octonionic multiplication. Nearly Kähler $6$-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group $\mathrm{G}_2$: the metric cone over a Riemannian $6$-manifold $M$ has holonomy contained in $\mathrm{G}_2$ if and only if $M$ is a nearly Kähler $6$-manifold.
A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler $6$-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the $6$-sphere and on the product of a pair of $3$-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions.
-
[Acharya:sine:cone] B. S. Acharya, F. Denef, C. Hofman, and N. Lambert, Freund–Rubin revisited, 2003.
@MISC{Acharya:sine:cone,
author = {Acharya, B. S. and Denef, F. and Hofman, C. and Lambert, N.},
arxiv = {hep-th/0308046},
title = {{F}reund--{R}ubin {r}evisited},
year = {2003},
} -
[Alekseevsky:Dotti:Ferraris]
D. Alekseevsky, I. Dotti, and C. Ferraris, "Homogeneous Ricci positive $5$-manifolds," Pacific J. Math., vol. 175, iss. 1, pp. 1-12, 1996.@ARTICLE{Alekseevsky:Dotti:Ferraris, mrkey = {1419469},
number = {1},
issn = {0030-8730},
author = {Alekseevsky, D. and Dotti, Isabel and Ferraris, C.},
mrclass = {53C30 (53C25)},
journal = {Pacific J. Math.},
zblnumber = {0865.53041},
volume = {175},
mrnumber = {1419469},
fjournal = {Pacific Journal of Mathematics},
mrreviewer = {McKenzie Y. Wang},
coden = {PJMAAI},
title = {Homogeneous {R}icci positive {$5$}-manifolds},
year = {1996},
pages = {1--12},
doi = {10.2140/pjm.1996.175.1},
} -
[NK:twistor:spaces] B. Alexandrov, G. Grantcharov, and S. Ivanov, "Curvature properties of twistor spaces of quaternionic Kähler manifolds," J. Geom., vol. 62, iss. 1-2, pp. 1-12, 1998.
@ARTICLE{NK:twistor:spaces, mrkey = {1631453},
number = {1-2},
issn = {0047-2468},
author = {Alexandrov, Bogdan and Grantcharov, Gueo and Ivanov, Stefan},
mrclass = {53C28 (53C26)},
doi = {10.1007/BF01237595},
journal = {J. Geom.},
zblnumber = {0911.53019},
volume = {62},
mrnumber = {1631453},
fjournal = {Journal of Geometry},
mrreviewer = {Rafael Herrera},
coden = {JGMAY3},
title = {Curvature properties of twistor spaces of quaternionic {K}ähler manifolds},
year = {1998},
pages = {1--12},
} -
[Bar] C. Bär, "Real Killing spinors and holonomy," Comm. Math. Phys., vol. 154, iss. 3, pp. 509-521, 1993.
@ARTICLE{Bar, mrkey = {1224089},
number = {3},
issn = {0010-3616},
author = {B{ä}r, Christian},
mrclass = {53C25 (58G30)},
journal = {Comm. Math. Phys.},
zblnumber = {0778.53037},
volume = {154},
mrnumber = {1224089},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Oussama Hijazi},
coden = {CMPHAY},
title = {Real {K}illing spinors and holonomy},
year = {1993},
pages = {509--521},
doi = {10.1007/BF02102106},
} -
[Bedulli:Vezzoni] L. Bedulli and L. Vezzoni, "Torsion of ${ SU}(2)$-structures and Ricci curvature in dimension 5," Differential Geom. Appl., vol. 27, iss. 1, pp. 85-99, 2009.
@ARTICLE{Bedulli:Vezzoni, mrkey = {2488990},
number = {1},
issn = {0926-2245},
author = {Bedulli, Lucio and Vezzoni, Luigi},
mrclass = {53C10 (53C25)},
doi = {10.1016/j.difgeo.2008.06.008},
journal = {Differential Geom. Appl.},
zblnumber = {1204.53018},
volume = {27},
mrnumber = {2488990},
fjournal = {Differential Geometry and its Applications},
mrreviewer = {Frederik Witt},
coden = {DGAPEO},
title = {Torsion of {${\rm SU}(2)$}-structures and {R}icci curvature in dimension 5},
year = {2009},
pages = {85--99},
} -
[Bilal:Metzger] A. Bilal and S. Metzger, "Compact weak $G_2$-manifolds with conical singularities," Nuclear Phys. B, vol. 663, iss. 1-2, pp. 343-364, 2003.
@ARTICLE{Bilal:Metzger, mrkey = {1986853},
number = {1-2},
issn = {0550-3213},
author = {Bilal, Adel and Metzger, Steffen},
mrclass = {53C29},
doi = {10.1016/S0550-3213(03)00388-2},
journal = {Nuclear Phys. B},
zblnumber = {1028.83031},
volume = {663},
mrnumber = {1986853},
fjournal = {Nuclear Physics. B},
mrreviewer = {Spiro Karigiannis},
coden = {NUPBBO},
title = {Compact weak {$G\sb 2$}-manifolds with conical singularities},
year = {2003},
pages = {343--364},
} -
[Birkhoff:Rota] G. Birkhoff and G. Rota, Ordinary Differential Equations, Fourth ed., New York: John Wiley & Sons, 1989.
@BOOK{Birkhoff:Rota, mrkey = {0972977},
author = {Birkhoff, Garrett and Rota, Gian-Carlo},
mrclass = {34-01},
edition = {Fourth},
isbn = {0-471-86003-4},
address = {New York},
publisher = {John Wiley \& Sons},
zblnumber = {0102.29901},
mrnumber = {0972977},
mrreviewer = {Michal {\v{C}}ver{\v{c}}ko},
title = {Ordinary Differential Equations},
year = {1989},
pages = {xii+399},
} -
[Bohm:Spheres] C. Böhm, "Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces," Invent. Math., vol. 134, iss. 1, pp. 145-176, 1998.
@ARTICLE{Bohm:Spheres, mrkey = {1646591},
number = {1},
issn = {0020-9910},
author = {B{ö}hm, Christoph},
mrclass = {53C25 (53C30)},
doi = {10.1007/s002220050261},
journal = {Invent. Math.},
zblnumber = {0965.53033},
volume = {134},
mrnumber = {1646591},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Megan M. Kerr},
coden = {INVMBH},
title = {Inhomogeneous {E}instein metrics on low-dimensional spheres and other low-dimensional spaces},
year = {1998},
pages = {145--176},
} -
[Bohm:Complete] C. Böhm, "Non-compact cohomogeneity one Einstein manifolds," Bull. Soc. Math. France, vol. 127, iss. 1, pp. 135-177, 1999.
@ARTICLE{Bohm:Complete, mrkey = {1700472},
number = {1},
issn = {0037-9484},
author = {B{ö}hm, Christoph},
mrclass = {53C25 (34C60)},
url = {http://www.numdam.org/item?id=BSMF_1999__127_1_135_0},
journal = {Bull. Soc. Math. France},
zblnumber = {0935.53021},
volume = {127},
mrnumber = {1700472},
fjournal = {Bulletin de la Société Mathématique de France},
mrreviewer = {Megan M. Kerr},
coden = {BSMFAA},
title = {Non-compact cohomogeneity one {E}instein manifolds},
year = {1999},
pages = {135--177},
} -
[BG:Sasaki] C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford: Oxford University Press, 2008.
@BOOK{BG:Sasaki, mrkey = {2382957},
author = {Boyer, Charles P. and Galicki, Krzysztof},
mrclass = {53C25 (14J45 32J27 53-02 57R30 57S25)},
series = {Oxford Math. Monogr.},
isbn = {978-0-19-856495-9},
address = {Oxford},
publisher = {Oxford University Press},
zblnumber = {1155.53002},
mrnumber = {2382957},
mrreviewer = {Andrew Swann},
title = {Sasakian Geometry},
year = {2008},
pages = {xii+613},
} -
[Bredon] G. E. Bredon, Introduction to Compact Transformation Groups, New York: Academic Press, 1972, vol. 46.
@BOOK{Bredon, mrkey = {0413144},
author = {Bredon, Glen E.},
mrclass = {57E15},
series = {Pure Appl. Math.},
address = {New York},
publisher = {Academic Press},
zblnumber = {0246.57017},
volume = {46},
mrnumber = {0413144},
mrreviewer = {P. Y. Wang},
title = {Introduction to Compact Transformation Groups},
year = {1972},
pages = {xiii+459},
} -
[Bryant:special:holonomy] R. L. Bryant, "Metrics with exceptional holonomy," Ann. of Math., vol. 126, iss. 3, pp. 525-576, 1987.
@ARTICLE{Bryant:special:holonomy, mrkey = {0916718},
number = {3},
issn = {0003-486X},
author = {Bryant, Robert L.},
mrclass = {53C25 (53C15 53C57)},
doi = {10.2307/1971360},
journal = {Ann. of Math.},
zblnumber = {0637.53042},
volume = {126},
mrnumber = {0916718},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Marisa Fern{á}ndez},
coden = {ANMAAH},
title = {Metrics with exceptional holonomy},
year = {1987},
pages = {525--576},
} -
[Bryant:almost:complex:6D] R. L. Bryant, "On the geometry of almost complex 6-manifolds," Asian J. Math., vol. 10, iss. 3, pp. 561-605, 2006.
@ARTICLE{Bryant:almost:complex:6D, mrkey = {2253159},
number = {3},
issn = {1093-6106},
author = {Bryant, Robert L.},
mrclass = {53C15 (32Q60)},
doi = {10.4310/AJM.2006.v10.n3.a4},
journal = {Asian J. Math.},
zblnumber = {1114.53026},
volume = {10},
mrnumber = {2253159},
fjournal = {The Asian Journal of Mathematics},
mrreviewer = {J. T. Davidov},
title = {On the geometry of almost complex 6-manifolds},
year = {2006},
pages = {561--605},
} -
[Bryant:Salamon] R. L. Bryant and S. M. Salamon, "On the construction of some complete metrics with exceptional holonomy," Duke Math. J., vol. 58, iss. 3, pp. 829-850, 1989.
@ARTICLE{Bryant:Salamon, mrkey = {1016448},
number = {3},
issn = {0012-7094},
author = {Bryant, Robert L. and Salamon, Simon M.},
mrclass = {53C25 (53C57)},
doi = {10.1215/S0012-7094-89-05839-0},
journal = {Duke Math. J.},
zblnumber = {0681.53021},
volume = {58},
mrnumber = {1016448},
fjournal = {Duke Mathematical Journal},
mrreviewer = {Krzysztof Galicki},
coden = {DUMJAO},
title = {On the construction of some complete metrics with exceptional holonomy},
year = {1989},
pages = {829--850},
} -
[Butruille] J. Butruille, "Classification des variétés approximativement kähleriennes homogènes," Ann. Global Anal. Geom., vol. 27, iss. 3, pp. 201-225, 2005.
@ARTICLE{Butruille, mrkey = {2158165},
number = {3},
issn = {0232-704X},
author = {Butruille, Jean-Baptiste},
mrclass = {53C25 (53C15 53C30)},
doi = {10.1007/s10455-005-1581-x},
journal = {Ann. Global Anal. Geom.},
zblnumber = {1079.53044},
volume = {27},
mrnumber = {2158165},
fjournal = {Annals of Global Analysis and Geometry},
mrreviewer = {Andrew Swann},
coden = {AGAGDV},
title = {Classification des variétés approximativement kähleriennes homogènes},
year = {2005},
pages = {201--225},
} -
[Calabi:TAMS] E. Calabi, "Construction and properties of some $6$-dimensional almost complex manifolds," Trans. Amer. Math. Soc., vol. 87, pp. 407-438, 1958.
@ARTICLE{Calabi:TAMS, mrkey = {0130698},
issn = {0002-9947},
author = {Calabi, Eugenio},
mrclass = {53.80 (57.60)},
doi = {10.2307/1993108},
journal = {Trans. Amer. Math. Soc.},
zblnumber = {0080.37601},
volume = {87},
mrnumber = {0130698},
fjournal = {Transactions of the American Mathematical Society},
title = {Construction and properties of some {$6$}-dimensional almost complex manifolds},
year = {1958},
pages = {407--438},
} -
[Candelas:delaOssa] P. Candelas and X. C. de la Ossa, "Comments on conifolds," Nuclear Phys. B, vol. 342, iss. 1, pp. 246-268, 1990.
@ARTICLE{Candelas:delaOssa, mrkey = {1068113},
number = {1},
issn = {0550-3213},
author = {Candelas, Philip and de la Ossa, Xenia C.},
mrclass = {32G13 (14J15 14J30 32G81 32J17 81T30)},
doi = {10.1016/0550-3213(90)90577-Z},
journal = {Nuclear Phys. B},
volume = {342},
mrnumber = {1068113},
fjournal = {Nuclear Physics. B},
mrreviewer = {Andrew A. Bytsenko},
coden = {NUPBBO},
title = {Comments on conifolds},
year = {1990},
pages = {246--268},
} -
[Cheeger:Colding:almost:rigidity] J. Cheeger and T. H. Colding, "Lower bounds on Ricci curvature and the almost rigidity of warped products," Ann. of Math., vol. 144, iss. 1, pp. 189-237, 1996.
@ARTICLE{Cheeger:Colding:almost:rigidity, mrkey = {1405949},
number = {1},
issn = {0003-486X},
author = {Cheeger, Jeff and Colding, Tobias H.},
mrclass = {53C21 (53C20 53C23)},
doi = {10.2307/2118589},
journal = {Ann. of Math.},
zblnumber = {0865.53037},
volume = {144},
mrnumber = {1405949},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Joseph E. Borzellino},
coden = {ANMAAH},
title = {Lower bounds on {R}icci curvature and the almost rigidity of warped products},
year = {1996},
pages = {189--237},
} -
[Conlon:Hein:Duke] R. J. Conlon and H. Hein, "Asymptotically conical Calabi-Yau manifolds, I," Duke Math. J., vol. 162, iss. 15, pp. 2855-2902, 2013.
@ARTICLE{Conlon:Hein:Duke, mrkey = {3161306},
number = {15},
issn = {0012-7094},
author = {Conlon, Ronan J. and Hein, Hans-Joachim},
mrclass = {53C25 (14J32 32W20)},
doi = {10.1215/00127094-2382452},
journal = {Duke Math. J.},
zblnumber = {1283.53045},
volume = {162},
mrnumber = {3161306},
fjournal = {Duke Mathematical Journal},
mrreviewer = {Matthew B. Stenzel},
title = {Asymptotically conical {C}alabi-{Y}au manifolds, {I}},
year = {2013},
pages = {2855--2902},
} -
[Conlon:Hein:III] R. J. Conlon and H. Hein, Asymptotically conical Calabi-Yau manifolds, III, 2014.
@MISC{Conlon:Hein:III,
author = {Conlon, Ronan J. and Hein, Hans-Joachim},
arxiv = {1405.7140},
title = {Asymptotically conical {C}alabi-{Y}au manifolds, {III}},
year = {2014},
} -
[Conti:SE] D. Conti, "Cohomogeneity one Einstein-Sasaki 5-manifolds," Comm. Math. Phys., vol. 274, iss. 3, pp. 751-774, 2007.
@ARTICLE{Conti:SE, mrkey = {2328911},
number = {3},
issn = {0010-3616},
author = {Conti, Diego},
mrclass = {53C25 (57S15 57S25)},
doi = {10.1007/s00220-007-0286-3},
journal = {Comm. Math. Phys.},
zblnumber = {1143.53041},
volume = {274},
mrnumber = {2328911},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Andrew Swann},
coden = {CMPHAY},
title = {Cohomogeneity one {E}instein-{S}asaki 5-manifolds},
year = {2007},
pages = {751--774},
} -
[Conti] D. Conti, "Embedding into manifolds with torsion," Math. Z., vol. 268, iss. 3-4, pp. 725-751, 2011.
@ARTICLE{Conti, mrkey = {2818726},
number = {3-4},
issn = {0025-5874},
author = {Conti, Diego},
mrclass = {53C10 (53C25 53C27 53C29 53C38)},
doi = {10.1007/s00209-010-0692-7},
journal = {Math. Z.},
zblnumber = {1232.53044},
volume = {268},
mrnumber = {2818726},
fjournal = {Mathematische Zeitschrift},
mrreviewer = {Andrew Swann},
coden = {MAZEAX},
title = {Embedding into manifolds with torsion},
year = {2011},
pages = {725--751},
} -
[Conti:Salamon] D. Conti and S. Salamon, "Generalized Killing spinors in dimension 5," Trans. Amer. Math. Soc., vol. 359, iss. 11, pp. 5319-5343, 2007.
@ARTICLE{Conti:Salamon, mrkey = {2327032},
number = {11},
issn = {0002-9947},
author = {Conti, Diego and Salamon, Simon},
mrclass = {53C27 (53C29 53C38 53C42)},
doi = {10.1090/S0002-9947-07-04307-3},
journal = {Trans. Amer. Math. Soc.},
zblnumber = {1130.53033},
volume = {359},
mrnumber = {2327032},
fjournal = {Transactions of the American Mathematical Society},
mrreviewer = {Simon G. Chiossi},
coden = {TAMTAM},
title = {Generalized {K}illing spinors in dimension 5},
year = {2007},
pages = {5319--5343},
} -
[Cortes:Vasquez] V. Cortés and J. J. Vásquez, "Locally homogeneous nearly Kähler manifolds," Ann. Global Anal. Geom., vol. 48, iss. 3, pp. 269-294, 2015.
@ARTICLE{Cortes:Vasquez, mrkey = {3396468},
number = {3},
issn = {0232-704X},
author = {Cort{é}s, V. and V{á}squez, J. J.},
mrclass = {53C55 (53C25 53C30)},
doi = {10.1007/s10455-015-9470-4},
journal = {Ann. Global Anal. Geom.},
zblnumber = {1326.53104},
volume = {48},
mrnumber = {3396468},
fjournal = {Annals of Global Analysis and Geometry},
title = {Locally homogeneous nearly {K}ähler manifolds},
year = {2015},
pages = {269--294},
} -
[Eells:Salamon:Twistor] J. Eells and S. Salamon, "Twistorial construction of harmonic maps of surfaces into four-manifolds," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 12, iss. 4, pp. 589-640 (1986), 1985.
@ARTICLE{Eells:Salamon:Twistor, mrkey = {0848842},
number = {4},
issn = {0391-173X},
author = {Eells, J. and Salamon, S.},
mrclass = {58E20 (32L05 53C42)},
url = {http://www.numdam.org/item?id=ASNSP_1985_4_12_4_589_0},
journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
zblnumber = {0627.58019},
volume = {12},
mrnumber = {0848842},
fjournal = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
mrreviewer = {Claude LeBrun},
coden = {PSNAAI},
title = {Twistorial construction of harmonic maps of surfaces into four-manifolds},
year = {1985},
pages = {589--640 (1986)},
} -
[Eschenburg:Comparison:hypersurfaces] J. -H. Eschenburg, "Comparison theorems and hypersurfaces," Manuscripta Math., vol. 59, iss. 3, pp. 295-323, 1987.
@ARTICLE{Eschenburg:Comparison:hypersurfaces, mrkey = {0909847},
number = {3},
issn = {0025-2611},
author = {Eschenburg, J.-H.},
mrclass = {53C20},
doi = {10.1007/BF01174796},
journal = {Manuscripta Math.},
zblnumber = {0642.53044},
volume = {59},
mrnumber = {0909847},
fjournal = {Manuscripta Mathematica},
mrreviewer = {Karsten Grove},
coden = {MSMHB2},
title = {Comparison theorems and hypersurfaces},
year = {1987},
pages = {295--323},
} -
[Eschenburg:Wang] J. -H. Eschenburg and M. Y. Wang, "The initial value problem for cohomogeneity one Einstein metrics," J. Geom. Anal., vol. 10, iss. 1, pp. 109-137, 2000.
@ARTICLE{Eschenburg:Wang, mrkey = {1758585},
number = {1},
issn = {1050-6926},
author = {Eschenburg, J.-H. and Wang, McKenzie Y.},
mrclass = {53C25 (34A12 34C60 34E05 53C30)},
doi = {10.1007/BF02921808},
journal = {J. Geom. Anal.},
zblnumber = {0992.53033},
volume = {10},
mrnumber = {1758585},
fjournal = {The Journal of Geometric Analysis},
mrreviewer = {Megan M. Kerr},
title = {The initial value problem for cohomogeneity one {E}instein metrics},
year = {2000},
pages = {109--137},
} -
[Fernandez:nearly:hypo] M. Fernández, S. Ivanov, V. Muñoz, and L. Ugarte, "Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities," J. Lond. Math. Soc., vol. 78, iss. 3, pp. 580-604, 2008.
@ARTICLE{Fernandez:nearly:hypo, mrkey = {2456893},
number = {3},
issn = {0024-6107},
author = {Fern{á}ndez, Marisa and Ivanov, Stefan and Mu{ñ}oz, Vicente and Ugarte, Luis},
mrclass = {53C15 (53C25)},
doi = {10.1112/jlms/jdn044},
journal = {J. Lond. Math. Soc.},
zblnumber = {1158.53018},
volume = {78},
mrnumber = {2456893},
fjournal = {Journal of the London Mathematical Society. Second Series},
mrreviewer = {Simon G. Chiossi},
title = {Nearly hypo structures and compact nearly {K}ähler 6-manifolds with conical singularities},
year = {2008},
pages = {580--604},
} -
[Ferus:Karcher] D. Ferus and H. Karcher, "Non-rotational minimal spheres and minimizing cones," Comment. Math. Helv., vol. 60, iss. 2, pp. 247-269, 1985.
@ARTICLE{Ferus:Karcher, mrkey = {0800005},
number = {2},
issn = {0010-2571},
author = {Ferus, Dirk and Karcher, Hermann},
mrclass = {53C42 (53A10)},
doi = {10.1007/BF02567412},
journal = {Comment. Math. Helv.},
zblnumber = {0566.53052},
volume = {60},
mrnumber = {0800005},
fjournal = {Commentarii Mathematici Helvetici},
mrreviewer = {Shigeo Akiba},
coden = {COMHAX},
title = {Non-rotational minimal spheres and minimizing cones},
year = {1985},
pages = {247--269},
} -
[gauntlett:SE] J. P. Gauntlett, D. Martelli, J. Sparks, and D. Waldram, "Sasaki-Einstein metrics on $S^2\times S^3$," Adv. Theor. Math. Phys., vol. 8, iss. 4, pp. 711-734, 2004.
@ARTICLE{gauntlett:SE, mrkey = {2141499},
number = {4},
issn = {1095-0761},
author = {Gauntlett, Jerome P. and Martelli, Dario and Sparks, James and Waldram, Daniel},
mrclass = {53C25 (53C80 83E50)},
url = {http://projecteuclid.org/euclid.atmp/1117750699},
journal = {Adv. Theor. Math. Phys.},
zblnumber = {1136.53317},
volume = {8},
mrnumber = {2141499},
fjournal = {Advances in Theoretical and Mathematical Physics},
title = {Sasaki-{E}instein metrics on {$S\sp 2\times S\sp 3$}},
year = {2004},
pages = {711--734},
} -
[Gray:1972] A. Gray, "Riemannian manifolds with geodesic symmetries of order $3$," J. Differential Geometry, vol. 7, pp. 343-369, 1972.
@ARTICLE{Gray:1972, mrkey = {0331281},
issn = {0022-040X},
author = {Gray, Alfred},
mrclass = {53C30},
url = {http://projecteuclid.org/euclid.jdg/1214431159},
journal = {J. Differential Geometry},
zblnumber = {0275.53026},
volume = {7},
mrnumber = {0331281},
fjournal = {Journal of Differential Geometry},
mrreviewer = {David Blair},
title = {Riemannian manifolds with geodesic symmetries of order {$3$}},
year = {1972},
pages = {343--369},
} -
[Gray:Math:Annalen] A. Gray, "The structure of nearly Kähler manifolds," Math. Ann., vol. 223, iss. 3, pp. 233-248, 1976.
@ARTICLE{Gray:Math:Annalen, mrkey = {0417965},
number = {3},
issn = {0025-5831},
author = {Gray, Alfred},
mrclass = {53B35},
doi = {10.1007/BF01360955},
journal = {Math. Ann.},
zblnumber = {0345.53019},
volume = {223},
mrnumber = {0417965},
fjournal = {Mathematische Annalen},
mrreviewer = {Makuto Matsumoto},
title = {The structure of nearly {K}ähler manifolds},
year = {1976},
pages = {233--248},
} -
[Gray:Hervella] A. Gray and L. M. Hervella, "The sixteen classes of almost Hermitian manifolds and their linear invariants," Ann. Mat. Pura Appl., vol. 123, pp. 35-58, 1980.
@ARTICLE{Gray:Hervella, mrkey = {0581924},
issn = {0003-4622},
author = {Gray, Alfred and Hervella, Luis M.},
mrclass = {53C15},
doi = {10.1007/BF01796539},
journal = {Ann. Mat. Pura Appl.},
zblnumber = {0444.53032},
volume = {123},
mrnumber = {0581924},
fjournal = {Annali di Matematica Pura ed Applicata. Serie Quarta},
mrreviewer = {Kouei Sekigawa},
title = {The sixteen classes of almost {H}ermitian manifolds and their linear invariants},
year = {1980},
pages = {35--58},
} -
[Grunewald] R. Grunewald, "Six-dimensional Riemannian manifolds with a real Killing spinor," Ann. Global Anal. Geom., vol. 8, iss. 1, pp. 43-59, 1990.
@ARTICLE{Grunewald, mrkey = {1075238},
number = {1},
issn = {0232-704X},
author = {Grunewald, Ralf},
mrclass = {58G25 (53C21 58G30)},
doi = {10.1007/BF00055017},
journal = {Ann. Global Anal. Geom.},
zblnumber = {0704.53050},
volume = {8},
mrnumber = {1075238},
fjournal = {Annals of Global Analysis and Geometry},
mrreviewer = {Oussama Hijazi},
coden = {ANAGDV},
title = {Six-dimensional {R}iemannian manifolds with a real {K}illing spinor},
year = {1990},
pages = {43--59},
} -
[Hitchin:Kahler:Twistor] N. Hitchin, "Kählerian twistor spaces," Proc. London Math. Soc., vol. 43, iss. 1, pp. 133-150, 1981.
@ARTICLE{Hitchin:Kahler:Twistor, mrkey = {0623721},
number = {1},
issn = {0024-6115},
author = {Hitchin, Nigel},
mrclass = {32C10 (35L25 53C55 81E99)},
doi = {10.1112/plms/s3-43.1.133},
journal = {Proc. London Math. Soc.},
zblnumber = {0474.14024},
volume = {43},
mrnumber = {0623721},
fjournal = {Proceedings of the London Mathematical Society. Third Series},
mrreviewer = {G. M. Khenkin},
coden = {PLMTAL},
title = {Kählerian twistor spaces},
year = {1981},
pages = {133--150},
} -
[Hitchin:stable:forms] N. Hitchin, "Stable forms and special metrics," in Global Differential Geometry: The Mathematical Legacy of Alfred Gray, RI: Amer. Math. Soc., Providence, 2001, vol. 288, pp. 70-89.
@INCOLLECTION{Hitchin:stable:forms, mrkey = {1871001},
author = {Hitchin, Nigel},
mrclass = {53C25 (53C10 53C29 58A10 58E11)},
series = {Contemp. Math.},
address = {RI},
publisher = {Amer. Math. Soc., Providence},
doi = {10.1090/conm/288/04818},
zblnumber = {1004.53034},
volume = {288},
mrnumber = {1871001},
booktitle = {Global Differential Geometry: The Mathematical Legacy of {A}lfred {G}ray},
mrreviewer = {Santiago R. Simanca},
venue = {{B}ilbao, 2000},
title = {Stable forms and special metrics},
pages = {70--89},
year = {2001},
} -
[LeBrun:Salamon] C. LeBrun and S. Salamon, "Strong rigidity of positive quaternion-Kähler manifolds," Invent. Math., vol. 118, iss. 1, pp. 109-132, 1994.
@ARTICLE{LeBrun:Salamon, mrkey = {1288469},
number = {1},
issn = {0020-9910},
author = {LeBrun, Claude and Salamon, Simon},
mrclass = {53C25 (32L25)},
doi = {10.1007/BF01231528},
journal = {Invent. Math.},
zblnumber = {0815.53078},
volume = {118},
mrnumber = {1288469},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Henrik Pedersen},
coden = {INVMBH},
title = {Strong rigidity of positive quaternion-{K}ähler manifolds},
year = {1994},
pages = {109--132},
} -
[Malgrange] B. Malgrange, "Sur les points singuliers des équations différentielles," Enseignement Math., vol. 20, pp. 147-176, 1974.
@ARTICLE{Malgrange, mrkey = {0368074},
issn = {0013-8584},
author = {Malgrange, Bernard},
mrclass = {58F05 (34A20)},
journal = {Enseignement Math.},
zblnumber = {0299.34011},
volume = {20},
mrnumber = {0368074},
fjournal = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
mrreviewer = {R. Gerard},
title = {Sur les points singuliers des équations différentielles},
year = {1974},
pages = {147--176},
} -
[Nagy:Product] P. Nagy, "Nearly Kähler geometry and Riemannian foliations," Asian J. Math., vol. 6, iss. 3, pp. 481-504, 2002.
@ARTICLE{Nagy:Product, mrkey = {1946344},
number = {3},
issn = {1093-6106},
author = {Nagy, Paul-Andi},
mrclass = {53C15 (53C12 53C28)},
doi = {10.4310/AJM.2002.v6.n3.a5},
journal = {Asian J. Math.},
zblnumber = {1041.53021},
volume = {6},
mrnumber = {1946344},
fjournal = {The Asian Journal of Mathematics},
mrreviewer = {Maria Falcitelli},
title = {Nearly {K}ähler geometry and {R}iemannian foliations},
year = {2002},
pages = {481--504},
} -
[Podesta:Spiro:I] F. Podestà and A. Spiro, "Six-dimensional nearly Kähler manifolds of cohomogeneity one," J. Geom. Phys., vol. 60, iss. 2, pp. 156-164, 2010.
@ARTICLE{Podesta:Spiro:I, mrkey = {2587385},
number = {2},
issn = {0393-0440},
author = {Podest{à},
Fabio and Spiro, Andrea},
mrclass = {53C25 (53C15 57S15)},
doi = {10.1016/j.geomphys.2009.09.008},
journal = {J. Geom. Phys.},
zblnumber = {1184.53074},
volume = {60},
mrnumber = {2587385},
fjournal = {Journal of Geometry and Physics},
mrreviewer = {Andrew Swann},
coden = {JGPHE5},
title = {Six-dimensional nearly {K}ähler manifolds of cohomogeneity one},
year = {2010},
pages = {156--164},
} -
[Podesta:Spiro:II] F. Podestà and A. Spiro, "Six-dimensional nearly Kähler manifolds of cohomogeneity one (II)," Comm. Math. Phys., vol. 312, iss. 2, pp. 477-500, 2012.
@ARTICLE{Podesta:Spiro:II, mrkey = {2917173},
number = {2},
issn = {0010-3616},
author = {Podest{à},
Fabio and Spiro, Andrea},
mrclass = {53C10 (22E50 53C25)},
doi = {10.1007/s00220-012-1482-3},
journal = {Comm. Math. Phys.},
zblnumber = {1262.53062},
volume = {312},
mrnumber = {2917173},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Andrew Swann},
coden = {CMPHAY},
title = {Six-dimensional nearly {K}ähler manifolds of cohomogeneity one ({II})},
year = {2012},
pages = {477--500},
} -
[Reyes:Carrion:thesis] R. Reyes Carrion, Some special geometries defined by Lie groups, 1993.
@MISC{Reyes:Carrion:thesis,
author = {Reyes Carrion, Ramon},
note = {{D}. {P}hil. thesis, {U}niversity of {O}xford, {T}rinity},
title = {Some special geometries defined by {L}ie groups},
year = {1993},
} -
[Salamon:twistor] S. Salamon, "Quaternionic Kähler manifolds," Invent. Math., vol. 67, iss. 1, pp. 143-171, 1982.
@ARTICLE{Salamon:twistor, mrkey = {0664330},
number = {1},
issn = {0020-9910},
author = {Salamon, Simon},
mrclass = {53C15 (53C55)},
doi = {10.1007/BF01393378},
journal = {Invent. Math.},
zblnumber = {0486.53048},
volume = {67},
mrnumber = {0664330},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Wilhelm Singhof},
coden = {INVMBH},
title = {Quaternionic {K}ähler manifolds},
year = {1982},
pages = {143--171},
} -
[Sparks:survey] J. Sparks, "Sasaki-Einstein manifolds," in Surveys in Differential Geometry. Volume XVI. Geometry of Special Holonomy and Related Topics, Somerville, MA: Int. Press, 2011, pp. 265-324.
@INCOLLECTION{Sparks:survey, mrkey = {2893680},
author = {Sparks, James},
mrclass = {53C25 (32Q20)},
address = {Somerville, MA},
publisher = {Int. Press},
doi = {10.4310/SDG.2011.v16.n1.a6},
zblnumber = {1256.53037},
mrnumber = {2893680},
booktitle = {Surveys in Differential Geometry. {V}olume {XVI}. {G}eometry of Special Holonomy and Related Topics},
mrreviewer = {Valentino Tosatti},
title = {Sasaki-{E}instein manifolds},
pages = {265--324},
year = {2011},
} -
[Steenrod:Book] N. Steenrod, The Topology of Fibre Bundles, Princeton, NJ: Princeton Univ. Press, 1951, vol. 14.
@BOOK{Steenrod:Book, mrkey = {0039258},
author = {Steenrod, Norman},
mrclass = {56.0X},
series = {Princeton Math. Ser.},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
zblnumber = {0942.55002},
volume = {14},
mrnumber = {0039258},
mrreviewer = {S. Chern},
title = {The {T}opology of {F}ibre {B}undles},
year = {1951},
pages = {viii+224},
} -
[Stenzel] M. B. Stenzel, "Ricci-flat metrics on the complexification of a compact rank one symmetric space," Manuscripta Math., vol. 80, iss. 2, pp. 151-163, 1993.
@ARTICLE{Stenzel, mrkey = {1233478},
number = {2},
issn = {0025-2611},
author = {Stenzel, Matthew B.},
mrclass = {32C17 (53C35 53C55)},
doi = {10.1007/BF03026543},
journal = {Manuscripta Math.},
zblnumber = {0811.53049},
volume = {80},
mrnumber = {1233478},
fjournal = {Manuscripta Mathematica},
mrreviewer = {Ji Sheng Na},
coden = {MSMHB2},
title = {Ricci-flat metrics on the complexification of a compact rank one symmetric space},
year = {1993},
pages = {151--163},
} -
[Wolf:1965] J. A. Wolf, "Complex homogeneous contact manifolds and quaternionic symmetric spaces," J. Math. Mech., vol. 14, pp. 1033-1047, 1965.
@ARTICLE{Wolf:1965, mrkey = {0185554},
author = {Wolf, Joseph A.},
mrclass = {53.66 (57.45)},
journal = {J. Math. Mech.},
zblnumber = {0141.38202},
volume = {14},
mrnumber = {0185554},
mrreviewer = {S. Kobayashi},
title = {Complex homogeneous contact manifolds and quaternionic symmetric spaces},
year = {1965},
pages = {1033--1047},
} -
[Gray:Wolf] J. A. Wolf and A. Gray, "Homogeneous spaces defined by Lie group automorphisms. II," J. Differential Geometry, vol. 2, pp. 115-159, 1968.
@ARTICLE{Gray:Wolf, mrkey = {0236329},
issn = {0022-040X},
author = {Wolf, Joseph A. and Gray, Alfred},
mrclass = {22.70 (53.00)},
url = {http://projecteuclid.org/euclid.jdg/1214501139},
journal = {J. Differential Geometry},
zblnumber = {0182.24702},
volume = {2},
mrnumber = {0236329},
fjournal = {Journal of Differential Geometry},
mrreviewer = {R. Steinberg},
title = {Homogeneous spaces defined by {L}ie group automorphisms. {II}},
year = {1968},
pages = {115--159},
}
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