Let G(n,m)be a real Grassmann manifold of alI n-dimensional subspaces in R ̄n+mFor two points p,q in G(n,m),we first find the equivalent condition that p,q are not conjugatealong any geodesic,then we prove that the geodesics connecting p and q are countable。we canalso obtain the indexformula λ(k_1…,k_n), which is the index ofthe geodesic numbered (ki,…,k_n)。Finally, according to Morse Theory,we can conclude that the space of the picecwise smooth pathsconnecting p and q is homotopic to a countable CW-complex if where the dimension ofthe ceIl numbered(k_1,…,k_n)is λ(k_1…,k_n)。