We define the exponential map at ${\displaystyle p\in M}$  by

${\displaystyle \exp _{p}:T_{p}M\supset B_{\epsilon }(0)\longrightarrow M,\quad vt\longmapsto \gamma _{p,v}(t),}$ 

where ${\displaystyle \gamma _{p,v}}$  is the unique geodesic with ${\displaystyle \gamma _{p,v}(0)=p}$  and tangent ${\displaystyle \gamma _{p,v}'(0)=v\in T_{p}M}$  and ${\displaystyle \epsilon }$  is chosen small enough so that for every ${\displaystyle t\in [0,1],vt\in B_{\epsilon }(0)\subset T_{p}M}$  the geodesic ${\displaystyle \gamma _{p,v}(t)}$  is defined. So, if ${\displaystyle M}$  is complete, then, by the Hopf–Rinow theorem, ${\displaystyle \exp _{p}}$  is defined on the whole tangent space.

Let ${\displaystyle \alpha :I\rightarrow T_{p}M}$  be a curve differentiable in ${\displaystyle T_{p}M}$  such that ${\displaystyle \alpha (0):=0}$  and ${\displaystyle \alpha '(0):=v}$ . Since ${\displaystyle T_{p}M\cong \mathbb {R} ^{n}}$ , it is clear that we can choose ${\displaystyle \alpha (t):=vt}$ . In this case, by the definition of the differential of the exponential in ${\displaystyle 0}$  applied over ${\displaystyle v}$ , we obtain:

${\displaystyle T_{0}\exp _{p}(v)={\frac {\mathrm {d} }{\mathrm {d} t}}{\Bigl (}\exp _{p}\circ \alpha (t){\Bigr )}{\Big \vert }_{t=0}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Bigl (}\exp _{p}(vt){\Bigr )}{\Big \vert }_{t=0}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Bigl (}\gamma _{p,v}(t){\Bigr )}{\Big \vert }_{t=0}=\gamma _{p,v}'(0)=v.}$ 

So (with the right identification ${\displaystyle T_{0}T_{p}M\cong T_{p}M}$ ) the differential of ${\displaystyle \exp _{p}}$  is the identity. By the implicit function theorem, ${\displaystyle \exp _{p}}$  is a diffeomorphism on a neighborhood of ${\displaystyle 0\in T_{p}M}$ . The Gauss Lemma now tells that ${\displaystyle \exp _{p}}$  is also a radial isometry.

Let ${\displaystyle p\in M}$ . In what follows, we make the identification ${\displaystyle T_{v}T_{p}M\cong T_{p}M\cong \mathbb {R} ^{n}}$ .

Gauss's Lemma states: Let ${\displaystyle v,w\in B_{\epsilon }(0)\subset T_{v}T_{p}M\cong T_{p}M}$  and ${\displaystyle M\ni q:=\exp _{p}(v)}$ . Then, ${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w)\rangle _{q}=\langle v,w\rangle _{p}.}$ 

For ${\displaystyle p\in M}$ , this lemma means that ${\displaystyle \exp _{p}}$  is a radial isometry in the following sense: let ${\displaystyle v\in B_{\epsilon }(0)}$ , i.e. such that ${\displaystyle \exp _{p}}$  is well defined. And let ${\displaystyle q:=\exp _{p}(v)\in M}$ . Then the exponential ${\displaystyle \exp _{p}}$  remains an isometry in ${\displaystyle q}$ , and, more generally, all along the geodesic ${\displaystyle \gamma }$  (in so far as ${\displaystyle \gamma _{p,v}(1)=\exp _{p}(v)}$  is well defined)! Then, radially, in all the directions permitted by the domain of definition of ${\displaystyle \exp _{p}}$ , it remains an isometry.

Recall that

${\displaystyle T_{v}\exp _{p}\colon T_{p}M\cong T_{v}T_{p}M\supset T_{v}B_{\epsilon }(0)\longrightarrow T_{\exp _{p}(v)}M.}$ 

We proceed in three steps:

• ${\displaystyle T_{v}\exp _{p}(v)=v}$  : let us construct a curve

${\displaystyle \alpha :\mathbb {R} \supset I\rightarrow T_{p}M}$  such that ${\displaystyle \alpha (0):=v\in T_{p}M}$  and ${\displaystyle \alpha '(0):=v\in T_{v}T_{p}M\cong T_{p}M}$ . Since ${\displaystyle T_{v}T_{p}M\cong T_{p}M\cong \mathbb {R} ^{n}}$ , we can put ${\displaystyle \alpha (t):=v(t+1)}$ . Therefore,

${\displaystyle T_{v}\exp _{p}(v)={\frac {\mathrm {d} }{\mathrm {d} t}}{\Bigl (}\exp _{p}\circ \alpha (t){\Bigr )}{\Big \vert }_{t=0}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Bigl (}\exp _{p}(tv){\Bigr )}{\Big \vert }_{t=1}=\Gamma (\gamma )_{p}^{\exp _{p}(v)}v=v,}$ 

where ${\displaystyle \Gamma }$  is the parallel transport operator and ${\displaystyle \gamma (t)=\exp _{p}(tv)}$ . The last equality is true because ${\displaystyle \gamma }$  is a geodesic, therefore ${\displaystyle \gamma '}$  is parallel.

Now let us calculate the scalar product ${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w)\rangle }$ .

We separate ${\displaystyle w}$  into a component ${\displaystyle w_{T}}$  parallel to ${\displaystyle v}$  and a component ${\displaystyle w_{N}}$  normal to ${\displaystyle v}$ . In particular, we put ${\displaystyle w_{T}:=av}$ , ${\displaystyle a\in \mathbb {R} }$ .

The preceding step implies directly:

${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w)\rangle =\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{T})\rangle +\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle }$ 

${\displaystyle =a\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(v)\rangle +\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle =\langle v,w_{T}\rangle +\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle .}$ 

We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have:

${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle =\langle v,w_{N}\rangle =0.}$ 

• ${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle =0}$  :

Let us define the curve

${\displaystyle \alpha \colon [-\epsilon ,\epsilon ]\times [0,1]\longrightarrow T_{p}M,\qquad (s,t)\longmapsto tv+tsw_{N}.}$ 

Note that

${\displaystyle \alpha (0,1)=v,\qquad {\frac {\partial \alpha }{\partial t}}(s,t)=v+sw_{N},\qquad {\frac {\partial \alpha }{\partial s}}(0,t)=tw_{N}.}$ 

Let us put:

${\displaystyle f\colon [-\epsilon ,\epsilon ]\times [0,1]\longrightarrow M,\qquad (s,t)\longmapsto \exp _{p}(tv+tsw_{N}),}$ 

and we calculate:

${\displaystyle T_{v}\exp _{p}(v)=T_{\alpha (0,1)}\exp _{p}\left({\frac {\partial \alpha }{\partial t}}(0,1)\right)={\frac {\partial }{\partial t}}{\Bigl (}\exp _{p}\circ \alpha (s,t){\Bigr )}{\Big \vert }_{t=1,s=0}={\frac {\partial f}{\partial t}}(0,1)}$ 

and

${\displaystyle T_{v}\exp _{p}(w_{N})=T_{\alpha (0,1)}\exp _{p}\left({\frac {\partial \alpha }{\partial s}}(0,1)\right)={\frac {\partial }{\partial s}}{\Bigl (}\exp _{p}\circ \alpha (s,t){\Bigr )}{\Big \vert }_{t=1,s=0}={\frac {\partial f}{\partial s}}(0,1).}$ 

Hence

${\displaystyle \langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w_{N})\rangle =\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial s}}\right\rangle (0,1).}$ 

We can now verify that this scalar product is actually independent of the variable ${\displaystyle t}$ , and therefore that, for example:

${\displaystyle \left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial s}}\right\rangle (0,1)=\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial s}}\right\rangle (0,0)=0,}$ 

because, according to what has been given above:

${\displaystyle \lim _{t\rightarrow 0}{\frac {\partial f}{\partial s}}(0,t)=\lim _{t\rightarrow 0}T_{tv}\exp _{p}(tw_{N})=0}$ 

being given that the differential is a linear map. This will therefore prove the lemma.

• We verify that ${\displaystyle {\frac {\partial }{\partial t}}\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial s}}\right\rangle =0}$ : this is a direct calculation. Since the maps ${\displaystyle t\mapsto f(s,t)}$  are geodesics,

${\displaystyle {\frac {\partial }{\partial t}}\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial s}}\right\rangle =\left\langle \underbrace {{\frac {D}{\partial t}}{\frac {\partial f}{\partial t}}} _{=0},{\frac {\partial f}{\partial s}}\right\rangle +\left\langle {\frac {\partial f}{\partial t}},{\frac {D}{\partial t}}{\frac {\partial f}{\partial s}}\right\rangle =\left\langle {\frac {\partial f}{\partial t}},{\frac {D}{\partial s}}{\frac {\partial f}{\partial t}}\right\rangle ={\frac {1}{2}}{\frac {\partial }{\partial s}}\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial t}}\right\rangle .}$ 

Since the maps ${\displaystyle t\mapsto f(s,t)}$  are geodesics, the function ${\displaystyle t\mapsto \left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial t}}\right\rangle }$  is constant. Thus,

${\displaystyle {\frac {\partial }{\partial s}}\left\langle {\frac {\partial f}{\partial t}},{\frac {\partial f}{\partial t}}\right\rangle ={\frac {\partial }{\partial s}}\left\langle v+sw_{N},v+sw_{N}\right\rangle =2\left\langle v,w_{N}\right\rangle =0.}$ 

• Riemannian geometry
• Metric tensor

• do Carmo, Manfredo (1992), Riemannian geometry, Basel, Boston, Berlin: Birkhäuser, ISBN 978-0-8176-3490-2