# Vocabulary Word

**Word:**manifold

**Definition:**many in number or kind; numerous; varied

## Sentences Containing 'manifold'

To look back upon things of former ages, as upon the manifold changes and conversions of several monarchies and commonwealths.

Whether it be a mere dissolution and unbinding of the manifold intricacies and entanglements of the confused atoms; or some such dispersion of the simple and incorruptible elements...

This if thou also shalt use to do, thou shalt rid thyself of that manifold luggage, wherewith thou art round about encumbered.

But I have heard, Mr. Holmes, that you can see deeply into the manifold wickedness of the human heart.

But these manifold mistakes in depicting the whale are not so very surprising after all.

This gives an equivalent more geometric way of describing the connection in terms of lifting paths in the manifold to paths in the frame bundle.

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold which is calculated from the metric tensor on a larger manifold into which the submanifold is embedded.

Let
be a map from the domain of the curve formula_7 with parameter formula_8 into the euclidean manifold formula_9.

In other words, the manifold formula_7 admits Riemannian metrics with higher systolic ratio formula_9 than for its symmetric metric, see Bangert et al.

In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable.

Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold.

Based on the Helmholtz resonance principle, the intake manifold is equipped with 3 chambers tuned to a specific resonant frequency.

A series of two butterfly valves coupled with electronically controlled actuators varied the volume and length of a resonant chamber within the intake manifold.

If "X" is a manifold, "Mf" will be a manifold of dimension one higher, and it is said to "fiber over the circle".

A deep result of Thurston states that in this case the 3-manifold "Mf" is hyperbolic if and only if "f" is a pseudo-Anosov homeomorphism of "S".

In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution.

A simple example of a statistical manifold, taken from physics, would be the canonical ensemble: it is a one-dimensional manifold, with the temperature "T" serving as the coordinate on the manifold.

Thus, the dosage is the coordinate on the manifold.

Let "X" be an orientable manifold, and let formula_1 be a measure on "X".

The statistical manifold "S"("X") of "X" is defined as the space of all measures formula_6 on "X" (with the sigma-algebra formula_7 held fixed).

Restricting to gauge transformations that preserve the connection at infinity gives a 4"k"-dimensional manifold "M""k", which is a circle bundle over the true moduli space and carries a natural complete hyperKähler metric (c.f. also Kähler–Einstein manifold).

With suspected to any of the complex structures of the hyper-Kähler family, this manifold is holomorphically equivalent to the space of based rational mapping of degree "k" from "P"1 to itself.

Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle.

If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position" it becomes an embedding.

In particular, a disc (of dimension 2) in general position will have no self intersections inside a manifold of dimension greater than 2+2.

If the manifold is 4 dimensional, this does not work: the problem is that a disc in general position may have double points where two points of the disc have the same image.

Informally we can think of this as taking a small neighborhood of the skeleton (thought of as embedded in some 4-manifold).

Monogenic functions are special cases of harmonic spinors on a spin manifold.

From an Atiyah–Singer–Dirac operator "D" we have the Lichnerowicz formula
where τ is the scalar curvature on the manifold, and Γ* is the adjoint of Γ.

If "M" is compact and τ ≥ 0 and τ > 0 somewhere then there are no non-trivial harmonic spinors on the manifold.

This allows us to note that over the space of smooth spinor sections the operator "D" is invertible such a manifold.

In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology.

Let "M" be a compact Riemannian manifold with positive sectional curvature.

The "boost" is the pressure to which the air–fuel mixture is compressed before being fed through to the engine's cylinders (manifold pressure).

This has complex dimension "n", but topological dimension, as a real manifold, 2"n", and is compact, connected, and orientable.

A nonsingular complex projective algebraic curve will then be a smooth orientable surface as a real manifold, embedded in a compact real manifold of dimension 2"n" which is CP"n" regarded as a real manifold.

A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions.

In mathematics, a space form is a complete Riemannian manifold "M" of constant sectional curvature "K".

In mathematics, particularly in differential topology, the preimage theorem is a theorem concerning the preimage of particular points in a manifold under the action of a smooth map.

This shows that the dimension is constant on a variety
This relies the dimension of a variety to that of a differentiable manifold.

More precisely, if "V" if defined over the reals, then the set of its real regular points is a differentiable manifold that has the same dimension as variety and as a manifold.

This is the algebraic analogue to the fact that a connected manifold has a constant dimension.

The DOHC 2.0 liter non turbocharged engine had a dual stage intake manifold.

As a result, the universal cover of any closed manifold "M" of constant negative curvature −1, which is to say, a hyperbolic manifold, is H"n".

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary.

Suppose first that "M" is an oriented Riemannian manifold.

The gradient of a scalar function ƒ is the vector field grad "f" that may be defined through the inner product formula_7 on the manifold, as
for all vectors "vx" anchored at point "x" in the tangent space "TxM" of the manifold at point "x".

On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic.

One expects the geometric properties of the equilibrium manifold to be related to the macroscopic physical properties.

The manifold formula_28 is naturally equipped
with the Riemannian metric formula_29.