Geodesic
Note on Almost Product Manifolds and their Tangent Bundles
Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 621-630

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Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P. 
Houh, Chorng Shi. Note on Almost Product Manifolds and their Tangent Bundles. Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 621-630. doi: 10.4153/CMB-1966-075-0
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     title = {Note on {Almost} {Product} {Manifolds} and their {Tangent} {Bundles}},
     journal = {Canadian mathematical bulletin},
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     year = {1966},
     volume = {9},
     number = {5},
     doi = {10.4153/CMB-1966-075-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-075-0/}
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