$2.$ Let S $$ RLet $Ssub{R}^3$ be an embedded $2-$dimensional manifold in $R^3.$ At each point pinS, let $N_p$ be the

set of all normal vectors to the surface. Define the disjoint union of all normal vectors,

$N$:$={}_{\text{p}\text{i}\text{n}\text{S}}{N}_p=\{(p,n_p)in(S,R^3)|n_p{?}_{|}{?}_T{?}_{p}S\},$

and note the natural projection map

$$:$NS,(p,n_p)|p.$

Consider a local $($embedding$)$ chart $$:$R^2$UsubS,$(u,v)|(u,v)$ and find the corresponding

basis for each $N_{(u,v)}.$ Using this define a bijection

tilde$()$:${}^{-1}(U)UR$

Consider the corresponding construction for an other local chart $$:$R^2$VsubS such that $UV$ is

non$-$empty. Calculate the transition function and argue that $$:$NS$ forms a line bundle over $S.$

$3$ be an embedded $2-$dimensional manifold in R

$3$

$.$ At each point p P S$,$ let Np be the

set of all normal vectors to the surface. Define the disjoint union of all normal vectors,

N :

$$

pPS

Np tpp$,$ npq P pS$,$ R

$3$

q $|$ np K TpSu $,$

and note the natural projection map

$\backslash$pi : N $$ S$,$ pp$,$ npq $$ p$.$

Consider a local $($embedding$)$ chart $\backslash$phi : R

$2$ U $$ S$,$ pu$,$ vq $\backslash$phi pu$,$ vq and find the corresponding

basis for each N$\backslash$phi pu$,$vq

$.$ Using this define a bijection

$\backslash$phi $$ : $\backslash$pi

$1$

pUq $$ U R

Consider the corresponding construction for an other local chart $\backslash$psi : R

$2$ V $$ S such that U XV is

non$-$empty. Calculate the transition function and argue that $\backslash$pi : N $$ S forms a line bundle over S