Let $S \subset \mathbb{R}^{3}$ be a surface and $p \in S$. Define the exponential map exp $p$ and compute its differential $\left.d \exp _{p}\right|_{0}$. Deduce that $\exp _{p}$ is a local diffeomorphism.

Give an example of a surface $S$ and a point $p \in S$ for which the exponential map $\exp _{p}$ fails to be defined globally on $T_{p} S$. Can this failure be remedied by extending the surface? In other words, for any such $S$, is there always a surface $S \subset \widehat{S} \subset \mathbb{R}^{3}$ such that the exponential map $\widehat{\exp }_{p}$ defined with respect to $\widehat{S}$is globally defined on $T_{p} S=T_{p} \widehat{S}$?

State the version of the Gauss-Bonnet theorem with boundary term for a surface $S \subset \mathbb{R}^{3}$ and a closed disc $D \subset S$ whose boundary $\partial D$ can be parametrized as a smooth closed curve in $S$.

Let $S \subset \mathbb{R}^{3}$ be a flat surface, i.e. $K=0$. Can there exist a closed disc $D \subset S$, whose boundary $\partial D$ can be parametrized as a smooth closed curve, and a surface $\tilde{S} \subset \mathbb{R}^{3}$ such that all of the following hold:

(i) $(S \backslash D) \cup \partial D \subset \tilde{S}$;

(ii) letting $\tilde{D}$ be $(\tilde{S} \backslash(S \backslash D)) \cup \partial D$, we have that $\tilde{D}$ is a closed disc in $\tilde{S}$ with boundary $\partial \tilde{D}=\partial D$

(iii) the Gaussian curvature $\tilde{K}$ of $\tilde{S}$ satisfies $\tilde{K} \geqslant 0$, and there exists a $p \in \tilde{S}$ such that $\tilde{K}(p)>0$ ?

Justify your answer.