In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, if there exists a function on $(M^n, g)$ such that its Hessian matrix is the identity matrix? Welcome some examples, thanks!

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, do there exists a function on $(M^n, g)$ such that its Hessian matrix is the identity matrix? Welcome some examples, thanks!

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. while on a smooth manifold $(M^n, g)$, if there exist a function on $(M^n, g)$ such that its Hessian matrix is an identity matrix? Welcome some examples, thanks!

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, if there exists a function on $(M^n, g)$ such that its Hessian matrix is the identity matrix? Welcome some examples, thanks!

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $1/2|x|^2$ is the identity matrix. while on a smooth manifold $(M^n, g)$, if there exsit a function on $(M^n, g)$ such that its Hessian matrix is an identity matrix? Welcome some examples, thanks!

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. while on a smooth manifold $(M^n, g)$, if there exist a function on $(M^n, g)$ such that its Hessian matrix is an identity matrix? Welcome some examples, thanks!