  In this paper, we construct the almost product structure of the minimal
Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline
Tate tensors, similar to Oort's and Mantovan's construction for Shimura
varieties of PEL-type. It allows us to describe the geometry of the Newton
stratum in terms of the geometry of two simpler objects, the central leaf and
the isogeny leaf. This yields the dimension and the closure relations of the
Newton strata in the deformation space. In particular, their nonemptiness shows
that a generalisation of Grothendieck's conjecture of deformations of
Barsotti-Tate groups with given Newton polygon holds.
  As an application, we determine analogous geometric properties of the Newton
stratification of Shimura varieties of Hodge type and prove the
equidimensionality of Rapoport-Zink spaces of Hodge type.
