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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 19th 2019

added statement of the definition

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 19th 2019

added statement of this fact:

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

$G = Sp(2) \times Sp(1) \simeq Spin(5) \times Spin(3) \hookrightarrow Spin(8)$

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

$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$
• CommentRowNumber3.
• CommentAuthorJames Francese
• CommentTimeMar 20th 2019

I’ve added a pointer to quaternionic manifold.

• CommentRowNumber4.
• CommentAuthorJames Francese
• CommentTimeMar 20th 2019

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

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

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

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 22nd 2019

added the original (?)

• 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 $X$ is given, but only under the assumption that

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

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

• 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)
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