  In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach
Spaces', we provided a geometrical description of manifolds of tensors in
Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces,
that allowed to extend the Dirac-Frenkel variational principle in the framework
of topological tensor spaces. The purpose of this note is to extend these
results to more general tensor formats. More precisely, we provide a new
geometrical description of manifolds of tensors in tree-based (or hierarchical)
format, also known as tree tensor networks, which are intersections of
manifolds of tensors in Tucker format associated with different partitions of
the set of dimensions. The proposed geometrical description of tensors in
tree-based format is compatible with the one of manifolds of tensors in Tucker
format.
