Based on Geodesic, show smooth function is constant
$begingroup$
Assume that a surface $S$ admits a parametrization $x(u,v)$ such that the first fundamental form with respect to this parametrization has the form
$(U(u) + V(v))(du^2 + dv^2)$
for some smooth functions $U$ and $V$ of one variable surfaces. Show that if $alpha(t) = x(alpha_1(t),alpha_2(t))$ is a geodesic on $S$, then
$U(alpha_1(t))sin^2varphi(t) - V(alpha_2(t))cos^2varphi(t)$
is constant, where $varphi(t)$ is an angle from $x_u(a_1(t), a_2(t))$ to $alpha'(t)$
--
I know from this I can conclude coefficients of the first fundamental form, but I don't know how I can relate it / what else I can conclude to show what the problem wants.
differential-geometry
asked Dec 4 '18 at 8:20
prestopresto
324
$endgroup$
add a comment |
$begingroup$
Assume that a surface $S$ admits a parametrization $x(u,v)$ such that the first fundamental form with respect to this parametrization has the form
$(U(u) + V(v))(du^2 + dv^2)$
for some smooth functions $U$ and $V$ of one variable surfaces. Show that if $alpha(t) = x(alpha_1(t),alpha_2(t))$ is a geodesic on $S$, then
$U(alpha_1(t))sin^2varphi(t) - V(alpha_2(t))cos^2varphi(t)$
is constant, where $varphi(t)$ is an angle from $x_u(a_1(t), a_2(t))$ to $alpha'(t)$
--
I know from this I can conclude coefficients of the first fundamental form, but I don't know how I can relate it / what else I can conclude to show what the problem wants.
differential-geometry
asked Dec 4 '18 at 8:20
prestopresto
324
$endgroup$
add a comment |
$begingroup$
Assume that a surface $S$ admits a parametrization $x(u,v)$ such that the first fundamental form with respect to this parametrization has the form
$(U(u) + V(v))(du^2 + dv^2)$
for some smooth functions $U$ and $V$ of one variable surfaces. Show that if $alpha(t) = x(alpha_1(t),alpha_2(t))$ is a geodesic on $S$, then
$U(alpha_1(t))sin^2varphi(t) - V(alpha_2(t))cos^2varphi(t)$
is constant, where $varphi(t)$ is an angle from $x_u(a_1(t), a_2(t))$ to $alpha'(t)$
--
I know from this I can conclude coefficients of the first fundamental form, but I don't know how I can relate it / what else I can conclude to show what the problem wants.
differential-geometry
asked Dec 4 '18 at 8:20
prestopresto
324
$endgroup$
Assume that a surface $S$ admits a parametrization $x(u,v)$ such that the first fundamental form with respect to this parametrization has the form
$(U(u) + V(v))(du^2 + dv^2)$
for some smooth functions $U$ and $V$ of one variable surfaces. Show that if $alpha(t) = x(alpha_1(t),alpha_2(t))$ is a geodesic on $S$, then
$U(alpha_1(t))sin^2varphi(t) - V(alpha_2(t))cos^2varphi(t)$
is constant, where $varphi(t)$ is an angle from $x_u(a_1(t), a_2(t))$ to $alpha'(t)$
--
I know from this I can conclude coefficients of the first fundamental form, but I don't know how I can relate it / what else I can conclude to show what the problem wants.
differential-geometry
differential-geometry
asked Dec 4 '18 at 8:20
prestopresto
324
asked Dec 4 '18 at 8:20
prestopresto
324
asked Dec 4 '18 at 8:20
prestopresto
324
asked Dec 4 '18 at 8:20
prestopresto
324
asked Dec 4 '18 at 8:20
prestopresto
324
324
add a comment |
add a comment |
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