Sasaki metric

Let ${\displaystyle (M,g)}$ be a Riemannian manifold, denote by ${\displaystyle \tau \colon \mathrm {T} M\to M}$ the tangent bundle over ${\displaystyle M}$. The Sasaki metric ${\displaystyle {\hat {g}}}$ on ${\displaystyle \mathrm {T} M}$ is uniquely defined by the following properties:

• The map ${\displaystyle \tau \colon \mathrm {T} M\to M}$ is a Riemannian submersion.
• The metric on each tangent space ${\displaystyle \mathrm {T} _{p}\subset \mathrm {T} M}$ is the Euclidean metric induced by ${\displaystyle g}$.
• Assume ${\displaystyle \gamma (t)}$ is a curve in ${\displaystyle M}$ and ${\displaystyle v(t)\in \mathrm {T} _{\gamma (t)}}$ is a parallel vector field along ${\displaystyle \gamma }$. Note that ${\displaystyle v(t)}$ forms a curve in ${\displaystyle \mathrm {T} M}$. For the Sasaki metric, we have ${\displaystyle v'(t)\perp \mathrm {T} _{\gamma (t)}}$for any ${\displaystyle t}$; that is, the curve ${\displaystyle v(t)}$ normally crosses the tangent spaces ${\displaystyle \mathrm {T} _{\gamma (t)}\subset \mathrm {T} M}$.

• S. Sasaki, On the differential geometry of tangent bundle of Riemannian manifolds, Tôhoku Math. J.,10 (1958), 338–354.