## 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 $p$.