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Fubini Study Geodesics on $Bbb CP^n$

Positive definiteness of Fubini-Study metricdeterminant of the Fubini-Study metricComputing the Fubini-Study metricFubini-Study norm of homogeneus polynomialsComputing the Fubini-Study metric in affine chart$omega_FS^n$, the top form of Fubini Study metricFubini-Study metric with respect to a specific coordinate systemFinding the Beltrami-Klein GeodesicsWhy is Fubini-Study metric invariant under $SU(n+1)$ action on $mathbbP^n mathbbC$Computing Fubini-Study metric from the formal definition

2

$begingroup$

I want to solve the geodesic equation on $mathbbCP^n$ with the Fubini Study metric
$$
g_ij=fracleft(1+mid zmid^2right)delta_ij-barz_iz_jleft(1+mid zmid^2right)^2
$$
I have worked out the Christoffel symbols for the above metric. They are as follows
$$
Gamma^k_ij=-fracdelta^k_jbarz_i+delta^k_ibarz_jleft(1+mid zmid^2right)
$$
Thus the affinely parametrized geodesic equation becomes
$$
fracd^2z^k(t)dt^2-frac2barz_j(t)left(1+mid z(t)mid^2right)fracdz^k(t)dtfracdz^j(t)dt=0
$$
I want to solve this equation for the case of $n=1,2$. I have never solved an ODE with complex variables.

Any suggestion or help is appreciated.

complex-analysis ordinary-differential-equations differential-geometry complex-geometry geodesic

share|cite|improve this question

edited Mar 22 at 16:58

m1rohit

asked Mar 22 at 9:35

m1rohitm1rohit

516

$endgroup$

add a comment | 

2

$begingroup$

I want to solve the geodesic equation on $mathbbCP^n$ with the Fubini Study metric
$$
g_ij=fracleft(1+mid zmid^2right)delta_ij-barz_iz_jleft(1+mid zmid^2right)^2
$$
I have worked out the Christoffel symbols for the above metric. They are as follows
$$
Gamma^k_ij=-fracdelta^k_jbarz_i+delta^k_ibarz_jleft(1+mid zmid^2right)
$$
Thus the affinely parametrized geodesic equation becomes
$$
fracd^2z^k(t)dt^2-frac2barz_j(t)left(1+mid z(t)mid^2right)fracdz^k(t)dtfracdz^j(t)dt=0
$$
I want to solve this equation for the case of $n=1,2$. I have never solved an ODE with complex variables.

Any suggestion or help is appreciated.

complex-analysis ordinary-differential-equations differential-geometry complex-geometry geodesic

share|cite|improve this question

edited Mar 22 at 16:58

m1rohit

asked Mar 22 at 9:35

m1rohitm1rohit

516

$endgroup$

add a comment | 

$begingroup$

I want to solve the geodesic equation on $mathbbCP^n$ with the Fubini Study metric
$$
g_ij=fracleft(1+mid zmid^2right)delta_ij-barz_iz_jleft(1+mid zmid^2right)^2
$$
I have worked out the Christoffel symbols for the above metric. They are as follows
$$
Gamma^k_ij=-fracdelta^k_jbarz_i+delta^k_ibarz_jleft(1+mid zmid^2right)
$$
Thus the affinely parametrized geodesic equation becomes
$$
fracd^2z^k(t)dt^2-frac2barz_j(t)left(1+mid z(t)mid^2right)fracdz^k(t)dtfracdz^j(t)dt=0
$$
I want to solve this equation for the case of $n=1,2$. I have never solved an ODE with complex variables.

Any suggestion or help is appreciated.

complex-analysis ordinary-differential-equations differential-geometry complex-geometry geodesic

share|cite|improve this question

edited Mar 22 at 16:58

m1rohit

asked Mar 22 at 9:35

m1rohitm1rohit

516

$endgroup$

share|cite|improve this question

edited Mar 22 at 16:58

m1rohit

asked Mar 22 at 9:35

m1rohitm1rohit

516

share|cite|improve this question

edited Mar 22 at 16:58

m1rohit

asked Mar 22 at 9:35

m1rohitm1rohit

516

asked Mar 22 at 9:35

m1rohitm1rohit

516

asked Mar 22 at 9:35

m1rohitm1rohit

516