In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
In the mathematical field of differential geometry, the Gauss–Bonnet theorem is a fundamental formula which links the curvature of a surface to its underlying topology.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because we could cut the handle and shrink it into the surface. But a Conway sphere is incompressible, because there are essential parts of a knot or link both inside and out, so there is no way to move the entire knot or link to one side of the punctured sphere. The mathematical definition is as follows. There are two cases to consider. A sphere is incompressible if both inside and outside the sphere there are some obstructions that prevent the sphere from shrinking to a point and also prevent the sphere from expanding to encompass all of space. A surface other than a sphere is incompressible if any disk with its boundary on the surface spans a disk in the surface.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product of the disk and the circle, endowed with the product topology.
In mathematics, tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation.
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993).
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor. Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.
In mathematics, Reeb sphere theorem, named after Georges Reeb, states that
In mathematics, Reeb stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the leaves are closed and have finite fundamental group.
In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Thurston boundary can be interpreted as the space of projective measured foliations on the surface.
In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete, this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.
Paul Alexander SchweitzerSJ is an American mathematician specializing in differential topology, geometric topology, and algebraic topology.
