Manifold

Technical description

In mathematics, a manifold is a topological space that looks locally like the "ordinary" Euclidean space Rn and is a Hausdorff space.

Related Topics:
Mathematics - Topological space - Euclidean space - Hausdorff space

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To make precise the notion of "looks locally like" one uses local coordinate systems or charts. A connected manifold has a definite topological dimension, which equals the number of coordinates needed in each local coordinate system. The foundational aspects of the subject were clarified during the 1930s, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory.

Related Topics:
Charts - Connected - 1930s - 19th century - Differential geometry - Lie group

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If the local charts on a manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold. These manifolds are called differentiable. In order to measure lengths and angles, even more structure is needed: one defines Riemannian manifolds to recover these geometrical ideas.

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