ALIBABA 2021 Problem 14

Let $f$ be a smooth function on $\mathbb{R}^n$ , denote by $G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}$ . Let $g$ be the restriction of the Euclidean metric on $G_f$ .

(1) Prove that $g$ is a complete metric.

(2) If there exists $\Lambda>0$ , such that $-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n$ , where $I_n$ is the unit matrix of order $n$ , and $\text{Hess}8f)$ is the Hessian matrix of $f$ , then the injectivity radius of $(G_f,g)$ is at least $\frac{\pi}{2\Lambda}$ .