Exponential map, Manifold

Question

For each $x\in M,X\in T_x M$ there exists a maximal interval $I_X\subseteq \mathbb{R}$ with $0\in I_X$ and a geodesic $c=c_X$ $$c:I\to M$$ with $c(0)=x,\dot{c}(0)=X$. Let $$C:=\{X\in TM | 1\in I_X\}$$ be a star-shaped neighborhood of the zero section of $TM$ and define an exponential map $$\exp:C\to M$$ $$X\mapsto c_X(1).$$ Now why if $$X\in C,0\leq t \leq1,$$ then $$\exp (tX)=c_X(t).$$ Also I do not know where were used the sandwiched inequalities for $t$,i.e $0\leq t \leq1$?

Answer 1

If $c(t)$ is geodesic then $\gamma(t)=c(kt)$ is also a geodesic. Thus given $Y=kX$ with $0<k<1$ we have that $\gamma(t)$ satisfies $\gamma(0)=x$ and $\gamma'(0)=kc'(0)=kX=Y.$ So, by definition, we have

$$exp_x(kX)=exp_x(Y)=\gamma(1)=c(k).$$

The condition $0\le k\le 1$ is used to guarantee that $Y\in C$ (in other words, to have that $exp_x(Y)$ is defined).