Any non-singular curve on a smooth surface will have its tangent vector T lying in the tangent plane of the surface orthogonal

* Any analytic function is smooth, that is, infinitely differentiable.

* Any smooth manifold is a diffeological space.

# 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 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 smooth function of one variable approximates a quadratic function when examined near enough to its minimum point ; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.

Any splices narrow enough to maintain smooth running would be less able to support the required weight.

The Bureau of Alcohol, Tobacco, Firearms & Explosives recognizes these firearms as being a smooth bore handgun which is an Any Other Weapon ( AOW ).

Any deviations from a smooth curve will cause the airflow over the leeward side of the sail to separate causing a reduction in lift and reduced performance.

Any such surface is the Kummer variety of the Jacobian of a smooth hyperelliptic curve of genus 2 ; i. e. a quotient of the Jacobian by the Kummer involution x ↦ − x.

Any number of caenophidian snakes can be induced to sidewind on artificial smooth surfaces, though difficulty in getting them to do so and their proficiency at it vary greatly.

The article continues: " Any lady must be as great an Infidel as the Grand Sultan himself, who, after receiving such authority can doubt that her skin will become as superlatively smooth, soft, white and delicate, as that of the lovely Fatima, whatever may have been its feel or its appearance before.

The canonical ring and therefore likewise the Kodaira dimension is a birational invariant: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings.

# 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 Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.

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 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.

Any timelike curve admits a comoving observer whose " time axis " corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis.

Any process involving virtual particles admits a schematic representation known as a Feynman diagram, which facilitates the understanding of calculations.

Any symplectic vector space admits a Darboux basis

Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation.

* Any uniformly continuous function admits a minimal modulus of continuity, that is sometimes referred to as the ( optimal ) modulus of continuity of:

Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold.

Any Riemannian symmetric space M is complete and Riemannian homogeneous ( meaning that the isometry group of M acts transitively on M ).

Any oriented Riemannian ( or pseudo-Riemannian ) manifold has a natural volume ( or pseudo volume ) form.

* Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space T < sub > p </ sub > H to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the " same " as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric

Any such sequence belongs to the Hilbert space ℓ < sub > 2 </ sub >, so the Hilbert cube inherits a metric from there.

Any metric tensor on a two-manifold can be written with respect to an exponential isothermal coordinate chart in the form

Any metric space has a smallest injective metric space into which it can be isometrically embedded, called its metric envelope or tight span.

* Any uniform space ( hence any metric space, topological vector space, or topological group ) is a Cauchy space ; see Cauchy filter for definitions.

Any compact Riemann surface of genus greater than 1 ( with its usual metric of constant curvature − 1 ) is a locally symmetric space but not a symmetric space.

Any ( pseudo ) metrizable space is uniformizable since the ( pseudo ) metric uniformity induces the ( pseudo ) metric topology.