What question did this study set out to answer?

The primary aim is to show that a specific product manifold admits a Riemannian metric with non-negative curvature.

January 17, 2026Proceedings of the American Mathematical Society0 citations

Non-negative curvature on the product manifold of 7-sphere and homotopy ℂℙ³

Key Points

The primary aim is to show that a specific product manifold admits a Riemannian metric with non-negative curvature.
Analyzed the first Pontrjagin class of the homotopy complex projective space.
Utilized properties of diffeomorphism to establish curvature conditions.
Demonstrated connections between curvature and manifold equivalency.
If the Pontrjagin class of the homotopy complex projective space is greater than or equal to 4, then curvature conditions are met.
Concluded that the product manifold of ℂℙ³ and S⁷ has a Riemannian metric of non-negative curvature.

Abstract

Let h C P 3 h CP³ denote a 6 6 -dimensional smooth manifold that is homotopy equivalent to the complex 3 3 -dimensional projective space C P 3 CP³. In this paper, we prove that if the first Pontrjagin class of h C P 3 h CP³ satisfies p 1 (h C P 3) ≥ 4 p₁ (h CP³) 4, then the product manifold h C P 3 × S 7 h CP³ S⁷ admits a Riemannian metric with non-negative sectional curvature. This conclusion follows from the diffeomorphism property: h C P 3 × S 7 h CP³ S⁷ is diffeomorphic to a certain manifold which admits a Riemannian metric with non-negative sectional curvature.

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