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# Disturbed natural forest Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.
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Any and manifold
# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.
Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.
* Any smooth manifold is a diffeological space.
Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system.
Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism a map with constant rank n.
Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.
Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.
* Any open manifold admits a ( non-complete ) Riemannian metric of positive ( or negative ) curvature.
Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.
Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.
* Any manifold with constant sectional curvature is an Einstein manifold — in particular:
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.
Any manifold is an n-handlebody, that is, any manifold is the union of handles.
Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot Carathéodory, defined as
Any compact manifold has isoperimetric dimension 0.
* Any compact symplectic manifold admits a symplectic Lefschetz pencil ( Donaldson 1999 ).
Any symplectic manifold ( or indeed any almost symplectic manifold ) has a natural volume form.

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