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For varities containing sufficiently many rational curves, the study collection of Hermitian symmetric spaces, there rational curves of minimal degree determine the global geometry to some extent. This collection is a geometric structure known as VMRT, the variety of minimal rational tangents. I'm not familiar with this area, but for certain varieties covered with lines (eg uniruled varieties), this collection or VMRTof . The rational curves contained in the variety can determine the geometry of the variety. In particular, the tagnent directions at passing through a general point can determine these rational curves. As a result, these curves which can be detected by algebraic geometry, is serve as an alternative to a replacement for the tangent space.

If the manifold is sufficiently variety has symmetry, for instance a Hermitian symmetric space, then the VMRT at a point tangent curves passing through two different points can be related canonically to compared. This will give the VMRT at another point, and this gives rise to geodesics and notion of curvature that if you are interested ask for. I am aware that Hwang and Mok has done work in this area.

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In the study of Hermitian symmetric spaces, there is a geometric structure known as VMRT, the variety of minimal rational tangents. I'm not familiar with this area, but for certain varieties covered with lines (eg uniruled varieties), this collection VMRT of rational curves contained in the variety can determine the geometry of the variety. In particular, the tagnent directions at a general point can determine these rational curves. As a result, these curves which can be detected by algebraic geometry, is a replacement for the tangent space.

If the manifold is sufficiently symmetric, then the VMRT at a point can be related canonically to the VMRT at another point, and this gives rise to geodesics and curvature that you are interested in.