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The complex Grassmannian SU(n)/S(U(k) * SU(n-k)) being a Hermitian symmetric space enjoys the property that its geodesics (in the standard Kaehler metric) are homogeneous, i.e., generated by action of a one parameter subgroup of SU(n). In the following reference there is an explicit construction of this map in the affine coordinates.

http://www.emis.de/journals/BBMS/Bulletin/bul972/berceanu1.pdf

Update:

Another method for the computation of the geodesics on symmetric spaces is through the solution of the radial part of the Hamilton-Jacobi equation. In the case of the complex Grassmannian, it depends on min(k, n-k) coordinates and depends only on the restricted roots of the symmetric space and their multiplicity (see, Helgason: Groups and geometric analysis for the definitions of the radial coordinates and the radial differential operators).

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The complex Grassmannian SU(n)/S(U(k) * SU(n-k)) being a Hermitaian Hermitian symmetric space enjoys the property that its geodesics (in the standard Kaehler metric) are homogeneous, i.e., generated by action of a one parameter subgroup of SU(n). In the following reference there is an explicit construction of this map in the affine coordinates.

http://www.emis.de/journals/BBMS/Bulletin/bul972/berceanu1.pdf

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The complex Grassmannian SU(n)/S(U(k) * SU(n-k)) being a Hermitaian symmetric space enjoys the property that its geodesics (in the standard Kaehler metric) are homogeneous, i.e., generated by action of a one parameter subgroup of SU(n). In the following reference there is an explicit construction of this map in the affine coordinates.

http://www.emis.de/journals/BBMS/Bulletin/bul972/berceanu1.pdf