  We introduce the equation of n-dimensional totally geodesic submanifolds of a
manifold E as a submanifold of the second order jet space of n-dimensional
submanifolds of E. Next we study the geometry of n-Grassmannian equivalent
connections, that is linear connections without torsion admitting the same
equation of n-dimensional totally geodesic submanifolds. We define the
n-Grassmannian structure as the equivalence class of such connections,
recovering for n=1 the case of theory of projectively equivalent connections.
By introducing the equation of parametrized n-dimensional totally geodesic
submanifolds as a submanifold of the second order jet space of the trivial
bundle on the space of parameters, we discover a relation of covering between
the `parametrized' equation and the `unparametrized' one. After having studied
symmetries of these equations, we discuss the case in which the space of
parameters is equal to R^n.

Differential Geometry;Analysis of PDEs