geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov. A closed manifold M is called...

coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise. Examples of essential manifolds include aspherical manifolds, real projective...

In mathematics, an analytic manifold, also known as a C ω {\displaystyle C^{\omega }} manifold, is a differentiable manifold with analytic transition maps...

manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...

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...

of multivariable calculus topics Manifold Differentiable manifold Smooth manifold Banach manifold Fréchet manifold Tensor analysis Tangent vector Tangent...

paper "Filling Riemannian manifolds"[G83] Gromov proved that every essential manifold M {\displaystyle M} with a Riemannian metric contains a closed non-contractible...

an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined...

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R 3 . {\displaystyle \mathbb {R} ^{3}.} J. H...

every essential surface is of area at least 1. Here a surface is called "essential" if it cannot be contracted to a point in the ambient 4-manifold. The...

common for symplectic manifolds satisfying only this weaker condition to be called "weakly exact." Acyclic space Essential manifold Whitehead conjecture...

Manifold Garden is a puzzle video game developed by American artist William Chyr. It was released on Windows, macOS, and iOS on October 18, 2019. The player...

mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally...

Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality...

the field is Gromov's inequality for the homotopy 1-systole of an essential n-manifold M: s y s π 1 ⁡ n ≤ C n vol ⁡ ( M ) , {\displaystyle \operatorname...

manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which...

Gromov's inequality (disambiguation) Gromov's systolic inequality for essential manifolds Systolic geometry Bangert, Victor; Katz, Mikhail G.; Shnider, Steve;...

systolic inequality for essential manifolds. To state his result, one requires a topological notion of an essential manifold. Similarly to Pu's inequality...

inequality for complex projective space Gromov's systolic inequality for essential manifolds Hadamard's inequality Hadwiger–Finsler inequality Hinge theorem Hitchin–Thorpe...

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the...

inequality for the real projective plane Gromov's systolic inequality for essential manifolds Gromov's inequality for complex projective space Eisenstein integer...

research, Guth has strengthened Gromov's systolic inequality for essential manifolds and, along with Nets Katz, found a solution to the Erdős distinct...

In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface...

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...

geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus,...

Lemma that in a reducible manifold every splitting is reducible. A Heegaard splitting is stabilized if there are essential simple closed curves α {\displaystyle...

of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures;...

piecewise-linear manifold and a topological manifold: a PL structure gives rise to a unique Lipschitz structure. While Lipschitz manifolds are closely related...

nation, Manifold joins S.W.O.R.D. as the 'Quintician', the leader of the Teleport Team. Destiny later learns that Manifold will be essential to the future...

of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein)...