  The main goal of this paper is to study the geometric structures associated
with the representation of tensors in subspace based formats. To do this we use
a property of the so-called minimal subspaces which allows us to describe the
tensor representation by means of a rooted tree. By using the tree structure
and the dimensions of the associated minimal subspaces, we introduce, in the
underlying algebraic tensor space, the set of tensors in a tree-based format
with either bounded or fixed tree-based rank. This class contains the Tucker
format and the Hierarchical Tucker format (including the Tensor Train format).
In particular, we show that the set of tensors in the tree-based format with
bounded (respectively, fixed) tree-based rank of an algebraic tensor product of
normed vector spaces is an analytic Banach manifold. Indeed, the manifold
geometry for the set of tensors with fixed tree-based rank is induced by a
fibre bundle structure and the manifold geometry for the set of tensors with
bounded tree-based rank is given by a finite union of connected components. In
order to describe the relationship between these manifolds and the natural
ambient space, we introduce the definition of topological tensor spaces in the
tree-based format. We prove under natural conditions that any tensor of the
topological tensor space under consideration admits best approximations in the
manifold of tensors in the tree-based format with bounded tree-based rank. In
this framework, we also show that the tangent (Banach) space at a given tensor
is a complemented subspace in the natural ambient tensor Banach space and hence
the set of tensors in the tree-based format with bounded (respectively, fixed)
tree-based rank is an immersed submanifold. This fact allows us to extend the
Dirac-Frenkel variational principle in the framework of topological tensor
spaces.
