Abstract
AbstractWe consider intermediate Ricci curvatures $$Ric_k$$
R
i
c
k
on a closed Riemannian manifold $$M^n$$
M
n
. These interpolate between the Ricci curvature when $$k=n-1$$
k
=
n
-
1
and the sectional curvature when $$k=1$$
k
=
1
. By establishing a surgery result for Riemannian metrics with $$Ric_k>0$$
R
i
c
k
>
0
, we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for $$Ric_k>0$$
R
i
c
k
>
0
when $$\lfloor n/2 \rfloor +2 \le k \le n-1.$$
⌊
n
/
2
⌋
+
2
≤
k
≤
n
-
1
.
This was previously known only in the case of positive Ricci curvature (Sha and Yang in J Differ Geom 29(1):95–103, 1989, J Differ Geom 33:127–138, 1991).
Funder
Deutsche Forschungsgemeinschaft
Karlsruhe House of Young Scientists
Publisher
Springer Science and Business Media LLC
Reference23 articles.
1. Amann, M., Quast, P., Zarei, M.: The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces. arXiv:2010.15742
2. Botvinnik, B., Walsh, M., Wraith, D.J.: Homotopy groups of the observer moduli space of Ricci positive metrics. Geom. Topol. 23, 3003–3040 (2019)
3. Chahine, J. K.: Manifolds with integral and intermediate Ricci curvature bounds, Ph.D. thesis, UC Santa Barbara (2019)
4. Crowley, D., Wraith, D.J.: Positive Ricci curvature on highly connected manifolds. J. Differ. Geom. 106, 187–243 (2017)
5. Domínguez-Vázquez, M., González-Álvaro, D., Mouillé, L.: Infinite families of manifolds of positive $$k^{th}$$-intermediate Ricci curvature with $$k$$ small. Math. Ann. 386, 1979–2014 (2022)
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献