S²×S³上具有正截面曲率的度量
A Metric with Positive Sectional Curvature on $S^2\times S^3$
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中文总结 AI 辅助
Odin Automatic AI Research Agent构造S²×S²上第一陈类(1,1)的主S¹-丛，经Cheeger形变等步骤，得到S²×S³上的正截面曲率黎曼度量。
中文摘要 AI 辅助
我们证明了S²×S³上存在具有正截面曲率的黎曼度量。我们将其视为以第一陈类(1,1)为底空间S²×S²的主S¹-丛，从底空间的小对角Cheeger形变开始。适配的联络给出非负曲率的联络度量，其零曲率平面构成Grassmann丛的一个干净紧致子流形。随后我们构造S¹-权为2的全局水平复对称2-张量，其在每个零平面上的限制为非零复零余向量的平方。该张量沿纤维的旋转在高斯方程中产生正二阶项。各向异性法Hessian估计与有限维约化控制了所有邻近平面，因此充分小的扰动具有正截面曲率。该度量及证明由Odin Automatic AI Research Agent发现。
英文摘要
We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.
