Eschenburg room

Eschenburg spaces are an important class of examples in the mathematical field of differential geometry . They are the simplest non-homogeneous examples of positively curved manifolds .

construction

The Eschenburg rooms are created as a biquotient of a left and right effect of the district group on the special unitary group . ${\displaystyle SU(3)}$

Let be and triples of integers with . Then we consider the two-sided effect of on the Lie group , which is also effective through left multiplication with the diagonal matrix and right multiplication . The biquotient of this effect is the Eschenburg area ${\displaystyle k=(k_{1},k_{2},k_{3})}$ ${\displaystyle l=(l_{1},l_{2},l_{3})}$ ${\displaystyle k_{1}+k_{2}+k_{3}=l_{1}+l_{2}+l_{3}}$ ${\displaystyle S^{1}=\left\{z\in \mathbb {C} \colon |z|=1\right\}}$ ${\displaystyle SU(3)}$ ${\displaystyle \operatorname {diag} (z^{k_{1}},z^{k_{2}},z^{k_{3}})}$ ${\displaystyle \operatorname {diag} (z^{l_{1}},z^{l_{2}},z^{l_{3}})}$

${\displaystyle E_{kl}=\operatorname {diag} (z^{k_{1}},z^{k_{2}},z^{k_{3}})\backslash SU(3)/\operatorname {diag} (z^{l_{1}},z^{l_{2}},z^{l_{3}})}$.

The effect is a free effect if and only if is not to be conjugated , i.e. if ${\displaystyle \operatorname {diag} (z^{k_{1}},z^{k_{2}},z^{k_{3}})}$ ${\displaystyle \operatorname {diag} (z^{l_{1}},z^{l_{2}},z^{l_{3}})}$

${\displaystyle kgV(k_{1}-l_{1};k_{2}-l_{2})=1,\ kgV(k_{1}-l_{2};k_{2}-l_{1})=1,\ kgV(k_{1}-l_{1};k_{2}-l_{3})=1,}$

${\displaystyle kgV(k_{1}-l_{2};k_{2}-l_{3})=1,\ kgV(k_{1}-l_{3};k_{2}-l_{1})=1,\ kgV(k_{1}-l_{3};k_{2}-l_{2})=1}$

applies.

For you get the Aloff Wallach rooms . ${\displaystyle l=(0,0,0)}$

properties

The metric induced by a certain left-invariant metric on has positive section curvature if and only if ${\displaystyle SU(3)}$ ${\displaystyle E_{kl}}$

${\displaystyle k_{i}\notin \left[\min(l_{1},l_{2},l_{3}),\ \max(l_{1},l_{2},l_{3})\right]}$ For ${\displaystyle i=1,2,3}$

applies.

There are a number of diffeomorphisms between Eschenburg spaces. Thus every permutation of the entries in or induces a diffeomorphic manifold. It applies and there is an (orientation-reversing) diffeomorphism between and . Furthermore, adding the same integer to all entries of and creates a diffeomorphic space. ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle E_{kl}\simeq E_{lk}}$ ${\displaystyle E_{kl}}$ ${\displaystyle E_{-k,-l}}$ ${\displaystyle k}$ ${\displaystyle l}$

The isometric group of an Eschenburg area has rank . ${\displaystyle 3}$

In particular, each Eschenburg room positive sectional curvature has a unique representation with ${\displaystyle E_{kl}}$

${\displaystyle k=(k_{1},k_{2},l_{1}+l_{2}-k_{1}-k_{2})}$

${\displaystyle l=(l_{1},l_{2},0)}$

${\displaystyle k_{1}\geq k_{2}>l_{1}\geq l_{2}\geq 0}$.

The following applies to the cohomology groups

${\displaystyle H^{1}(E_{kl})=0,H^{2}(E_{kl})=\mathbb {Z} }$

${\displaystyle H^{3}(E_{kl})=0,H^{4}(E_{kl})=\mathbb {Z} /r\mathbb {Z} }$

with . The producer of is the square of the producer of . ${\displaystyle r=\vert k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}-l_{1}l_{2}-l_{1}l_{3}-l_{2}l_{3}\vert }$ ${\displaystyle H^{4}}$ ${\displaystyle H^{2}}$

literature

• J.-H. Eschenburg : New examples of manifolds with strictly positive curvature , Invent. Math. 66, 469-480 (1982)
• K. Shankar: Strong inhomogeneity of Eschenburg spaces , Mich. Math. J. 50, 125-141 (2002)
• L. Astor, E. Micha, G. Pastor: On the homotopy type of Eschenburg spaces with positive sectional curvature , Proc. AMS 132, 3725-3729 (2004)
• B. Krüggel: Homeomorphism and diffeomorphism classification of Eschenburg spaces , Quart. J. Math. 56, 553-577 (2005)
• K. Grove , K. Shankar, W. Ziller : Symmetries of Eschenburg spaces and the Chern problem , Asian J. Math. 10, 647-661 (2006)
• T. Chinburg, C. Escher, W. Ziller: Topological properties of Eschenburg spaces and 3-Sasakian manifolds , Math. Ann. 339, 3-20 (2007)

Individual evidence

1. ↑ Eschenburg, op. Cit.
2. ↑ Grove-Shankar-Ziller, op. Cit.