Proposition: If a Riemannian manifold $(M,g)$ has non-positive sectional curvature, then $\forall p\in M,\mathrm{conj}(p)=\emptyset$. Proof: Suppose $q\in\mathrm{conj}(p)$ is …

早上在证明黎曼流形上的极值原理的时候意识到了这件事，于是把完整过程写一下以作记录。 Proposition: For a smooth map $f:M\to N$, the representation matrix of its derivative $df\in\Gamma(M,T^*M\otimes f^*TN)$ is orthogona…