; Isotropic quadratic form: A quadratic form q is said to be isotropic if there is a non-zero vector v such that q ( v )= 0.

; Isotropic coordinates on an Isotropic chart for Lorentzian manifolds.

A stronger assumption is necessary ; in dimensions four and higher there are simply connected manifolds which are not homeomorphic to an n-sphere.

Generalizations of the Gauss – Bonnet theorem to n-dimensional Riemannian manifolds were found in the 1940s, by Allendoerfer, Weil, and Chern ; see generalized Gauss – Bonnet theorem and Chern – Weil homomorphism.

Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian ; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.

We will consider only the case of orientable Haken manifolds, as this simplifies the discussion ; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a " thickened up " version of the surface, i. e. a trivial I-bundle.

In addition, Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem ; this is also true for virtually Haken manifolds.

* Boundary ( topology ), the closure minus the interior of a subset of a topological space ; an edge in the topology of manifolds, as in the case of a ' manifold with boundary '

Here is a short list of global results concerning manifolds with positive Ricci curvature ; see also classical theorems of Riemannian geometry.

Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston ; the possibilities, called Thurston model geometries, include the three-sphere S < sup > 3 </ sup >, three-dimensional Euclidean space E < sup > 3 </ sup >, three-dimensional hyperbolic space H < sup > 3 </ sup >, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds ; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

Most symplectic manifolds, one can say, are not Kähler ; and so do not have an integrable complex structure compatible with the symplectic form.

The most important of these shapes are so-called Calabi-Yau manifolds ; when the extra dimensions take on those particular forms, physics in three dimensions exhibits an abstract symmetry known as supersymmetry.

Characteristic classes were later found for foliations of manifolds ; they have ( in a modified sense, for foliations with some allowed singularities ) a classifying space theory in homotopy theory.

As a mathematician he is known for a number of contributions: the Cartan – Kähler theorem on singular solutions of non-linear analytic differential systems ; the idea of a Kähler metric on complex manifolds ; and the Kähler differentials, which provide a purely algebraic theory and have generally been adopted in algebraic geometry.

Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds ; see contact geometry.

More generally, one can also join manifolds together along identical submanifolds ; this generalization is often called the fiber sum.

A connected sum along a codimension-two can also be carried out in the category of symplectic manifolds ; this elaboration is called the symplectic sum.

Suppose that φ: M → N is a smooth map between smooth manifolds M and N ; then there is an associated linear map from the space of 1-forms on N ( the linear space of sections of the cotangent bundle ) to the space of 1-forms on M. This linear map is known as the pullback ( by φ ), and is frequently denoted by φ < sup >*</ sup >.

With the development of homology theory to include K-theory and other extraordinary theories from about 1955, it was realised that the homology H < sub >*</ sub > could be replaced by other theories, once the products on manifolds were constructed ; and there are now textbook treatments in generality.

Applications of K-groups were found from 1960 onwards in surgery theory for manifolds, in particular ; and numerous other connections with classical algebraic problems were brought out.

Some had been there for years ; ;

Some of the poems express a mood of joy in a newly discovered love ; ;

Some few and myself withstand his inhabitation and town privileges, without confession and reformation of his uncivil and inhuman practices at Portsmouth ; ;

Some would say that they were not permitted to run their businesses only for profit ; ;

Some departments will attack their new goals enthusiastically ; ;

Some manufacturers have had the foresight to provide a socket for the chuck key ; ;

Some men have no talent for or interest in management ; ;

Some are failing to achieve as much as their ability would permit ; ;

Some time in 1912, Picasso cut out and folded a piece of paper in the shape of a guitar ; ;

Some of the lime that is always on hand in the Capitol basement for plaster repairs was slaked several months for us ; ;

Some recent writings assume that the ignorant young couples are a thing of the remote, Victorian past ; ;

Some Catholic priests lecture there ; ;

Some persons are so sensitive to this truth as to propose that we do away with institutions altogether ; ;

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

We are thirsty and hungry ; ;

There are of course many Souths ; ;

Old attitudes are held more tenaciously in the Tidewater than the Piedmont ; ;

Both concepts are undergoing alteration ; ;

Often, too, the social institutions are housed in these pavilions and palaces and bridges, for these great structures are not simply `` historical monuments '' ; ;

the miraculous way in which music, revelation and death are associated in a single instant -- all this seems a triumph of art, a rather desperate art, in itself ; ;

He is a widower, his three children are dead, he has no one left on earth ; ;

His unsuccessful strivings to give up drink are represented as religious strivings ; ;

The images themselves, like their counterparts in experience, are not neutral qualities to be surveyed dispassionately ; ;

The most primitive feelings are rudimentary value feelings, both positive and negative: a desire to appropriate this or that part of the environment into oneself ; ;

The state universities of Maine, New Hampshire, And Vermont are older and more `` respectable '' ; ;