For a Riemannian manifold (N,g), we construct a scalar flat neutral metric G on the tangent bundle TN. The metric is locally conformally flat if and only if either N is a 2-dimensional manifold or (N,g) is a real space form. It is also shown that G is locally symmetric if and only if g is locally symmetric. We then study submanifolds in TN and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph in the tangent bundle TN to have parallel mean curvature are studied. Finally, using the cross product in R3 we show that the space of oriented lines in R3 can be minimally isometrically embedded in TR3.
- Almost para Kaehler structure
- Neutral metric
- Tangent bundle