manifold

# tag · @ author · ! scientist

Imagine an ant crawling on the surface of a huge ball. To it, the ball seems flat, like an endless plain. But in fact, it lives on a curved surface that looks flat up close. So a manifold is a space that in each small region looks like ordinary Euclidean space, but as a whole may be more complex, like the surface of the Earth or a donut.

History

The idea of manifolds originated in the works of Bernhard Riemann in the 19th century, but the modern language of topology was given to them by Henri Poincaré, who formulated the famous Poincaré conjecture about three-dimensional manifolds in the early 20th century.

How it works

Manifolds appear in physics: according to general relativity, spacetime is a four-dimensional manifold curved by mass and energy. Just as a stretched fabric sags under a heavy ball, a star causes spacetime to curve, creating gravity.

💡 If we lived on the surface of a donut, we could walk in any direction and eventually return to the starting point, unaware that our space has a hole! Manifolds allow us to study such unusual shapes of the universe.

A manifold is a topological space locally homeomorphic to Euclidean space ℝ^n. The topology of a manifold is given by an atlas of compatible charts. A smooth manifold is additionally endowed with a differentiable structure, allowing the use of calculus on it. Manifolds are classified by dimension, orientability, presence of boundaries, and other features. They serve as the main stage for many branches of mathematics and physics: from Riemannian geometry and Lie group theory to general relativity and gauge field theories.

History

The concept of a manifold arose from attempts to mathematically describe complex geometric shapes. Bernhard Riemann, in his 1854 lecture 'On the Hypotheses Which Lie at the Foundations of Geometry', introduced a generalized concept of multidimensional space. At the end of the 19th century, works by Felix Klein and Sophus Lie gave rise to the theory of Lie groups — smooth manifolds with a group structure. In the early 20th century, Henri Poincaré laid the foundations of topology, in particular, formulating the conjecture about the homotopy equivalence of the three-dimensional sphere. Later, the development of manifold topology led to the classification of surfaces, the creation of knot theory, and modern differential geometry.

How it works

In physics, spacetime is modeled as a four-dimensional pseudo-Riemannian manifold, whose metric is determined by Einstein's field equations: R_{μν} - ½Rg_{μν} = (8πG/c⁴) T_{μν}. Here, the manifold sets the geometric properties of the universe, and curvature describes gravitational interaction. In quantum mechanics, wave functions are defined on infinite-dimensional Hilbert spaces, but their configuration spaces are often finite-dimensional manifolds (e.g., the Bloch sphere for a single qubit). Gauge symmetry groups (U(1), SU(2), SU(3)) are Lie groups naturally acting on corresponding manifolds.

Open problems & fun facts

Classification of four-dimensional topological manifolds, including the smooth structures conjecture; The problem of constructing a quantum theory of gravity on a dynamical manifold background; Study of symplectic manifolds in the context of mirror symmetry and string theory

💡 The Poincaré conjecture, proved only in 2003 by Grigori Perelman, was one of the 'Millennium Problems'. Interestingly, Perelman himself declined the million-dollar prize and the Fields Medal, modestly noting that he simply completed Richard Hamilton's work.

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manifold

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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