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nLab > Latest Changes: quaternion-Kähler manifold

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2019

    added statement of the definition

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2019

    added statement of this fact:

    Let XX be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

    G=Sp(2)×Sp(1)≃Spin(5)×Spin(3)↪Spin(8) G = Sp(2) \times Sp(1) \simeq Spin(5) \times Spin(3) \hookrightarrow Spin(8)

    then the Euler class χ\chi, first Pontryagin class p 1p_1 and second Pontryagin class p 2p_2 of the frame bundle/tangent bundle are related by

    8χ=4p 2−(p 1) 2. 8 \chi \;=\; 4 p_2 - (p_1)^2 \,.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorJames Francese
    • CommentTimeMar 20th 2019

    I’ve added a pointer to quaternionic manifold.

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorJames Francese
    • CommentTimeMar 20th 2019

    Adding several references ranging from book-level to special topic articles

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2019

    Thanks!

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2019
    • (edited Mar 21st 2019)

    added a few references:

    general exposition

    on the case with positive scalar curvature:

    • Amann, Positive Quaternion Kähler Manifolds (pdf)

    • Amann, Partial Classification Results for Positive Quaternion Kaehler Manifolds (arXiv:0911.4587)

    and this one

    • Claude LeBrun, On complete quaternionic-Kähler manifolds, Duke Math. J. Volume 63, Number 3 (1991), 723-743 (euclid:1077296077)

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2019
    • (edited Mar 21st 2019)

    copied over James Example “qK is q” also to here

    diff, v12, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2019

    added the original (?)

    diff, v16, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 11th 2019

    In 8.1-8.3 of CV98 a cohomological characterization of existence of quaternion-Kaehler structure on an 8-manifold XX is given, but only under the assumption that

    H 2(X,ℤ/2)=0. H^2(X,\mathbb{Z}/2)=0 \,.

    I’d like to get a better feeling for this assumption. Does it rule out “a lot” of qK-manifolds, or is it “harmless”?

    Sorry, very vague question.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2019

    started adding statements about positive quaternion-Kaehler manifolds: here

    diff, v21, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2019
    • (edited Apr 13th 2019)

    .

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2019

    finally added pointer to

    • Y. S. Poon, Simon Salamon, Quaternionic Kähler 8-manifolds with positive scalar curvature, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 (euclid:1214446322)

    diff, v23, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2020

    added pointer to

    • Edmond Bonan, Sur les GG-structures de type quaternionien, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 9 (1967) no. 4, p. 389-463 (numdam:CTGDC_1967__9_4_389_0)

    for discussion in terms of G-structure

    diff, v25, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2020

    also

    diff, v25, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2020

    added pointer to:

    • K. Galicki, H. Blaine Lawson, Quaternionic Reduction and Quaternionic Orbifolds, Mathematische Annalen (1988) Volume: 282, Issue: 1, page 1-22 (dml:164446)

    diff, v26, current

1 to 15 of 15

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