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

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal h double-struck upper C double-struck upper P cubed"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {h}\mathbb {CP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6"> <mml:semantics> <mml:mn>6</mml:mn> <mml:annotation encoding="application/x-tex">6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional smooth manifold that is homotopy equivalent to the complex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional projective space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C double-struck upper P cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {CP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we prove that if the first Pontrjagin class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal h double-struck upper C double-struck upper P cubed"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {h}\mathbb {CP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German p 1 left-parenthesis normal h double-struck upper C double-struck upper P cubed right-parenthesis greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak p_1(\mathrm {h}\mathbb {CP}^3)\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the product manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal h double-struck upper C double-struck upper P cubed times upper S Superscript 7"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>7</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {h}\mathbb {CP}^3\times S^7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a Riemannian metric with non-negative sectional curvature. This conclusion follows from the diffeomorphism property: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal h double-struck upper C double-struck upper P cubed times upper S Superscript 7"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>7</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {h}\mathbb {CP}^3\times S^7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is diffeomorphic to a certain manifold which admits a Riemannian metric with non-negative sectional curvature.