  We study the geometry of the tangent bundle equipped with a two-parameter
family of Riemannian metrics. After deriving the expression of the Levi-Civita
connection, we compute the Riemann curvature tensor and the sectional, Ricci
and scalar curvatures. Specializing to the case of space forms, we characterise
the metrics giving positive sectional curvature and show that one can always
find parameters ensuring positive scalar curvature on the tangent space. Under
some curvature conditions, this extends to general base manifolds.
