with
for
a function on the tangent bundle. In addition, is homogeneous of degree 1 in and of the form
(Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry.
Riemann introduced the ideas underlying the subject in an 1854 habilitation lecture that was published posthumously (Riemann 1868).
Euclidean geometry is the flat special case, while the surface of a sphere with its induced metric
gives a standard positively curved example. The subject provides the language used
in general relativity and many geometric models in physics (Big Think 2026).
for 
a function on the tangent bundle 
. In addition, 
is homogeneous of degree 1 in 
and of the form
