Using Line Element to find out length of a curve on a circle
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
$endgroup$
add a comment |
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
$endgroup$
add a comment |
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
$endgroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
5219
5219
add a comment |
add a comment |
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