Quaternionic Kähler manifold

In mathematics , quaternionic Kähler manifolds are a research area of differential geometry .

definition

A coherent , orientable Riemannian manifold of dimension is a quaternionic Kahler manifold if its holonomy in is included. In the case one also demands that it is a self-dual Einstein manifold . ${\displaystyle 4n}$ ${\displaystyle Sp(n)Sp(1)}$ ${\displaystyle n=1}$

Here the (compact) symplectic group is called and acts through left multiplication of and right multiplication (as diagonal matrices) of , which is understood as a subgroup of . ${\displaystyle Sp(n)}$ ${\displaystyle Sp(n)Sp(1)}$ ${\displaystyle \mathbb {H} ^{n}=\mathbb {R} ^{4n}}$ ${\displaystyle Sp(n)}$ ${\displaystyle Sp(1)}$ ${\displaystyle Sp(n)Sp(1)}$ ${\displaystyle GL(4n,\mathbb {R} )}$

A quaternionic Kähler manifold is called positive or negative if the Riemannian metric is complete and has positive or negative scalar curvature .

properties

• A quaternionic Kähler manifold is hypercaler if and only if its scalar curvature vanishes.

• Every positive quaternionic Kähler manifold is compact and simply connected .

• The only quaternionic Kähler manifold with positive intersection curvature is the quaternionic projective space .

All known examples of positive quaternionic Kähler manifolds are Wolf spaces ; The LeBrun-Salamon conjecture says that all positive quaternionic Kähler manifolds are symmetric spaces and thus (according to the classification of symmetric spaces) in particular Wolf spaces. (For n = 1 the conjecture of Hitchin and for n = 2 of Poon-Salamon was proven.)

Twistor room

For each quaternionic Kähler manifold one associates a so-called “twistor space” as follows. is overlaid by two and locally the bundle can be lifted into a bundle. The effect on can then be used to locally define an associated quaternionic line bundle . Even if this does not have to be defined globally, its complex projectivization is defined globally and you get a bundle ${\displaystyle Sp(n)Sp(1)}$ ${\displaystyle Sp(n)\times Sp(1)}$ ${\displaystyle Sp(n)Sp(1)}$ ${\displaystyle Sp(n)\times Sp(1)}$ ${\displaystyle Sp(1)}$ ${\displaystyle \mathbb {H} ^{1}=\mathbb {C} ^{2}}$ ${\displaystyle H}$

${\displaystyle \mathbb {C} P^{1}\to P_{\mathbb {C} }(H)\to M}$.

The space is called the twistor space of the quaternionic Kahler manifold . ${\displaystyle Z:=P_{\mathbb {C} }(H)}$ ${\displaystyle M}$

Example: The twistor space of the quaternionic-projective space is the complex-projective space and the bundle ${\displaystyle \mathbb {H} P^{n}}$ ${\displaystyle \mathbb {C} P^{2n+1}}$

${\displaystyle \mathbb {C} P^{1}\to \mathbb {C} P^{2n+1}\to \mathbb {H} P^{n}}$

is the canonical projection mapping.

Theorem (LeBrun-Salamon) : The twistor space of a positive quaternionic Kähler manifold is a Fano contact manifold , also compact, simply connected, Kählersch and Einsteinsch .

Furthermore, a positive quaternionic Kähler manifold is a symmetric space if and only if its twistor space is a homogeneous space (among biholomorphic maps ) .

literature

• Salamon, Simon: Quaternionic Kähler manifolds. Invent. Math. 67 (1982) no. 1, 143-171.
• Poon, YS; Salamon, SM: Quaternionic Kähler 8-manifolds with positive scalar curvature. J. Differential Geom. 33 (1991) no. 2, 363-378.
• LeBrun, Claude; Salamon, Simon: Strong rigidity of positive quaternion-Kähler manifolds. Invent. Math. 118 (1994) no. 1, 109-132.
• Salamon, Simon: Quaternionic Kähler Geometry. Proceedings of the University of Cambridge VI, 1999, 83-121.