New G$_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

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.

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  • [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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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},
      }

Authors

Lorenzo Foscolo

Stony Brook University, Stony Brook, NY

Mark Haskins

Imperial College London, South Kensington Campus, London, UK