Some geometric facts of Cylinder

Let S^1\times \mathbb{R}^1\subset \mathbb{R}^3  be a cylinder with radius 1.

\displaystyle \begin{cases}x=\cos\theta\\ y=\sin\theta\\ z=z\end{cases}

Pull back the metric of \mathbb{R}^3 to cylinder ds^2=d\theta^2+dz^2. Then cylinder is locally flat Riemanian manifold. Its curvature tensor is 0 and thus the sectional curvature is also 0.

\textbf{Thm:} If M has non-positive sectional curvature, then any point x\in M does not have conjugate points.

Suppose p\in M. Then Conj(p)=M. Actually, any two points can be connected by infinitely many geodesics. But the Seg(p) is the cylinder except a line apposite to p.

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