## Abstract

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive-definiteness of the curvature operator and positivity of sectional curvature, and stems from the work of Thorpe (J Differ Geom 5:113–125, 1971; Erratum. J Differ Geom 11:315, 1976). We prove that this condition is preserved under Riemannian submersions and Cheeger deformations and that most compact homogeneous spaces with positive sectional curvature satisfy it.

## Keywords

Riemannian geometry Algebraic curvature operators Riemannian submersion Positive sectional curvature Homogeneous spaces## Mathematics Subject Classification

53B20 53C20 53C21 53C30 53C35## Notes

### Acknowledgements

It is a pleasure to thank Karsten Grove, Thomas Püttmann, Luigi Verdiani and Wolfgang Ziller for their constant interest in this project and many valuable suggestions. We also thank Amy Buchmann, David Johnson, and Martin Kerin for helpful conversations on related subjects.

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