Geodesic
Selfadjoint Metrics on Almost Tangent Manifolds Whose Riemannian Connection is Almost Tangent
Canadian mathematical bulletin, Tome 17 (1975) no. 5, pp. 671-674

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Let M be a differentiable manifold of class C ∞, with a given (1, 1) tensor field J of constant rank such that J2=λI (for some real constant λ). J defines a class of conjugate (G-structures on M. For λ>0, one particular representative structure is an almost product structure. Almost complex structure arises when λ<0. If the rank of J is maximum and λ=0, then we obtain an almost tangent structure. In the last two cases the dimension of the manifold is necessarily even. A Riemannian metric S on M is said to be related if one of the conjugate structures defined by S has a common subordinate structure with the G-structure defined by S. It is said to be J-metric if the orthogonal structure defined by S has a common subordinate structure. 
Goel, D. S. Selfadjoint Metrics on Almost Tangent Manifolds Whose Riemannian Connection is Almost Tangent. Canadian mathematical bulletin, Tome 17 (1975) no. 5, pp. 671-674. doi: 10.4153/CMB-1974-121-2
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     author = {Goel, D. S.},
     title = {Selfadjoint {Metrics} on {Almost} {Tangent} {Manifolds} {Whose} {Riemannian} {Connection} is {Almost} {Tangent}},
     journal = {Canadian mathematical bulletin},
     pages = {671--674},
     year = {1975},
     volume = {17},
     number = {5},
     doi = {10.4153/CMB-1974-121-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-121-2/}
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