Minimizing properties of geodesics problem in do Carmo's book

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I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.
enter image description here
I don't understand why $langlefracpartial f partial r, fracpartial f partial t rangle=0$. Can someone fill in the details?



This is the Gauss lemma that he is talking about.
enter image description here
So my question becomes, how did he apply this lemma in order to obtain that inner product equal to zero.




differential-geometry riemannian-geometry geodesic




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    Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
    – John Ma
    Jul 30 at 14:51










  • some discussion here
    – John Ma
    Jul 30 at 14:52






  • 1




    @JohnMa Thanks!! this is what I will do in the future!
    – Hurjui Ionut
    Jul 30 at 15:22














up vote
1
down vote

favorite












I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.
enter image description here
I don't understand why $langlefracpartial f partial r, fracpartial f partial t rangle=0$. Can someone fill in the details?



This is the Gauss lemma that he is talking about.
enter image description here
So my question becomes, how did he apply this lemma in order to obtain that inner product equal to zero.




differential-geometry riemannian-geometry geodesic




share|cite|improve this question

















  • 1




    Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
    – John Ma
    Jul 30 at 14:51










  • some discussion here
    – John Ma
    Jul 30 at 14:52






  • 1




    @JohnMa Thanks!! this is what I will do in the future!
    – Hurjui Ionut
    Jul 30 at 15:22












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.
enter image description here
I don't understand why $langlefracpartial f partial r, fracpartial f partial t rangle=0$. Can someone fill in the details?



This is the Gauss lemma that he is talking about.
enter image description here
So my question becomes, how did he apply this lemma in order to obtain that inner product equal to zero.




differential-geometry riemannian-geometry geodesic




share|cite|improve this question













I'm DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.
enter image description here
I don't understand why $langlefracpartial f partial r, fracpartial f partial t rangle=0$. Can someone fill in the details?



This is the Gauss lemma that he is talking about.
enter image description here
So my question becomes, how did he apply this lemma in order to obtain that inner product equal to zero.





differential-geometry riemannian-geometry geodesic





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 31 at 10:43









John Ma

37.5k93669




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asked Jul 30 at 14:40









Hurjui Ionut

314111




314111







  • 1




    Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
    – John Ma
    Jul 30 at 14:51










  • some discussion here
    – John Ma
    Jul 30 at 14:52






  • 1




    @JohnMa Thanks!! this is what I will do in the future!
    – Hurjui Ionut
    Jul 30 at 15:22












  • 1




    Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
    – John Ma
    Jul 30 at 14:51










  • some discussion here
    – John Ma
    Jul 30 at 14:52






  • 1




    @JohnMa Thanks!! this is what I will do in the future!
    – Hurjui Ionut
    Jul 30 at 15:22







1




1




Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
– John Ma
Jul 30 at 14:51




Hi, you can actually edit your previous question instead of asking a new one (this would bump the question to the first page again). I read your comment two days ago but don't have time to write up an answer.
– John Ma
Jul 30 at 14:51












some discussion here
– John Ma
Jul 30 at 14:52




some discussion here
– John Ma
Jul 30 at 14:52




1




1




@JohnMa Thanks!! this is what I will do in the future!
– Hurjui Ionut
Jul 30 at 15:22




@JohnMa Thanks!! this is what I will do in the future!
– Hurjui Ionut
Jul 30 at 15:22










1 Answer
1






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1
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accepted










Note that $f(r,t) = exp(rv(t))$, hence by the chain rule$$
partial_r f(r_0,t_0)=(dexp_p)_r_0v(t_0)[v(t_0)]
$$
and
$$
partial _t f(r_0,t_0)=(dexp_p)_r_0v(t_0)[r_0dot v(t_0)].
$$
Hence
$$
langle partial_r f(r_0,t_0)vert partial_t f(r_0,t_0)rangle = langle (dexp_p)_r_0v(t_0)[v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle\ =r_0^-1langle (dexp_p)_r_0v(t_0)[r_0v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle oversettextGauß= r_0^-1 langle r_0 v(t_0) vert r_0dot v(t_0) rangle.
$$
The latter is zero as it is a multiple to the derivative of $tmapsto langle v(t)vert v(t) rangle equiv 1$.






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  • This was very clear! Thanks!
    – Hurjui Ionut
    Jul 30 at 15:24










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










Note that $f(r,t) = exp(rv(t))$, hence by the chain rule$$
partial_r f(r_0,t_0)=(dexp_p)_r_0v(t_0)[v(t_0)]
$$
and
$$
partial _t f(r_0,t_0)=(dexp_p)_r_0v(t_0)[r_0dot v(t_0)].
$$
Hence
$$
langle partial_r f(r_0,t_0)vert partial_t f(r_0,t_0)rangle = langle (dexp_p)_r_0v(t_0)[v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle\ =r_0^-1langle (dexp_p)_r_0v(t_0)[r_0v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle oversettextGauß= r_0^-1 langle r_0 v(t_0) vert r_0dot v(t_0) rangle.
$$
The latter is zero as it is a multiple to the derivative of $tmapsto langle v(t)vert v(t) rangle equiv 1$.






share|cite|improve this answer





















  • This was very clear! Thanks!
    – Hurjui Ionut
    Jul 30 at 15:24














up vote
1
down vote



accepted










Note that $f(r,t) = exp(rv(t))$, hence by the chain rule$$
partial_r f(r_0,t_0)=(dexp_p)_r_0v(t_0)[v(t_0)]
$$
and
$$
partial _t f(r_0,t_0)=(dexp_p)_r_0v(t_0)[r_0dot v(t_0)].
$$
Hence
$$
langle partial_r f(r_0,t_0)vert partial_t f(r_0,t_0)rangle = langle (dexp_p)_r_0v(t_0)[v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle\ =r_0^-1langle (dexp_p)_r_0v(t_0)[r_0v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle oversettextGauß= r_0^-1 langle r_0 v(t_0) vert r_0dot v(t_0) rangle.
$$
The latter is zero as it is a multiple to the derivative of $tmapsto langle v(t)vert v(t) rangle equiv 1$.






share|cite|improve this answer





















  • This was very clear! Thanks!
    – Hurjui Ionut
    Jul 30 at 15:24












up vote
1
down vote



accepted







up vote
1
down vote



accepted






Note that $f(r,t) = exp(rv(t))$, hence by the chain rule$$
partial_r f(r_0,t_0)=(dexp_p)_r_0v(t_0)[v(t_0)]
$$
and
$$
partial _t f(r_0,t_0)=(dexp_p)_r_0v(t_0)[r_0dot v(t_0)].
$$
Hence
$$
langle partial_r f(r_0,t_0)vert partial_t f(r_0,t_0)rangle = langle (dexp_p)_r_0v(t_0)[v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle\ =r_0^-1langle (dexp_p)_r_0v(t_0)[r_0v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle oversettextGauß= r_0^-1 langle r_0 v(t_0) vert r_0dot v(t_0) rangle.
$$
The latter is zero as it is a multiple to the derivative of $tmapsto langle v(t)vert v(t) rangle equiv 1$.






share|cite|improve this answer













Note that $f(r,t) = exp(rv(t))$, hence by the chain rule$$
partial_r f(r_0,t_0)=(dexp_p)_r_0v(t_0)[v(t_0)]
$$
and
$$
partial _t f(r_0,t_0)=(dexp_p)_r_0v(t_0)[r_0dot v(t_0)].
$$
Hence
$$
langle partial_r f(r_0,t_0)vert partial_t f(r_0,t_0)rangle = langle (dexp_p)_r_0v(t_0)[v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle\ =r_0^-1langle (dexp_p)_r_0v(t_0)[r_0v(t_0)]~vert~ (dexp_p)_r_0v(t_0)[r_0dot v(t_0)]rangle oversettextGauß= r_0^-1 langle r_0 v(t_0) vert r_0dot v(t_0) rangle.
$$
The latter is zero as it is a multiple to the derivative of $tmapsto langle v(t)vert v(t) rangle equiv 1$.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 30 at 15:05









Jan Bohr

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3,1241419











  • This was very clear! Thanks!
    – Hurjui Ionut
    Jul 30 at 15:24
















  • This was very clear! Thanks!
    – Hurjui Ionut
    Jul 30 at 15:24















This was very clear! Thanks!
– Hurjui Ionut
Jul 30 at 15:24




This was very clear! Thanks!
– Hurjui Ionut
Jul 30 at 15:24












 

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