Existence of a geodesic connecting two points p,q in a manifold M with intrinsic distance smaller than inj(p;M)

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Assume that $M$ is a smooth manifold and $r>0$ is smaller than the injectivity radius $operatornameinj(p;M)$ of a point $p in M$.



I really don't understand why the image of the set $v in T_pM mid $
under the exponential map must be given by $ q in M mid d(p,q) =r $
where $d$ denote the intrinsic distance of M (the infimum of the length under all smooth function connecting p to q).



I know that with the Gauss-Lemma one can show that
beginequation
exp_p(v in T_pM mid ) subseteq q in M mid d(p,q) =r ,
endequation
but what about the reverse inclusion? That is to say, given a point $q in M$ with $d(p,q) = r <inj(p;M)$ why does exist a vector $v in T_pM$
of length r such that $exp_p(v)=q$?




differential-geometry riemannian-geometry smooth-manifolds geodesic




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    up vote
    0
    down vote

    favorite












    Assume that $M$ is a smooth manifold and $r>0$ is smaller than the injectivity radius $operatornameinj(p;M)$ of a point $p in M$.



    I really don't understand why the image of the set $v in T_pM mid $
    under the exponential map must be given by $ q in M mid d(p,q) =r $
    where $d$ denote the intrinsic distance of M (the infimum of the length under all smooth function connecting p to q).



    I know that with the Gauss-Lemma one can show that
    beginequation
    exp_p(v in T_pM mid ) subseteq q in M mid d(p,q) =r ,
    endequation
    but what about the reverse inclusion? That is to say, given a point $q in M$ with $d(p,q) = r <inj(p;M)$ why does exist a vector $v in T_pM$
    of length r such that $exp_p(v)=q$?




    differential-geometry riemannian-geometry smooth-manifolds geodesic




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Assume that $M$ is a smooth manifold and $r>0$ is smaller than the injectivity radius $operatornameinj(p;M)$ of a point $p in M$.



      I really don't understand why the image of the set $v in T_pM mid $
      under the exponential map must be given by $ q in M mid d(p,q) =r $
      where $d$ denote the intrinsic distance of M (the infimum of the length under all smooth function connecting p to q).



      I know that with the Gauss-Lemma one can show that
      beginequation
      exp_p(v in T_pM mid ) subseteq q in M mid d(p,q) =r ,
      endequation
      but what about the reverse inclusion? That is to say, given a point $q in M$ with $d(p,q) = r <inj(p;M)$ why does exist a vector $v in T_pM$
      of length r such that $exp_p(v)=q$?




      differential-geometry riemannian-geometry smooth-manifolds geodesic




      share|cite|improve this question











      Assume that $M$ is a smooth manifold and $r>0$ is smaller than the injectivity radius $operatornameinj(p;M)$ of a point $p in M$.



      I really don't understand why the image of the set $v in T_pM mid $
      under the exponential map must be given by $ q in M mid d(p,q) =r $
      where $d$ denote the intrinsic distance of M (the infimum of the length under all smooth function connecting p to q).



      I know that with the Gauss-Lemma one can show that
      beginequation
      exp_p(v in T_pM mid ) subseteq q in M mid d(p,q) =r ,
      endequation
      but what about the reverse inclusion? That is to say, given a point $q in M$ with $d(p,q) = r <inj(p;M)$ why does exist a vector $v in T_pM$
      of length r such that $exp_p(v)=q$?





      differential-geometry riemannian-geometry smooth-manifolds geodesic





      share|cite|improve this question










      share|cite|improve this question




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      asked Jul 29 at 8:15









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