Definition of "manifold" []

• Many and varied; of many kinds; multiple: our manifold failings. (adjective)
• Having many features or forms: manifold intelligence. (adjective)
• Being such for a variety of reasons: a manifold traitor. (adjective)
• Consisting of or operating several devices of one kind at the same time. (adjective)
• A whole composed of diverse elements. (noun)

American Heritage(R) Dictionary of the English Language, Fifth Edition. Copyright (c) 2011 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

• "Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positive-definite metric tensor, d (x, y) ≥ 0."
• "A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3-space, for example)."
• "The Lorentzian manifold is a pseudo-Riemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positive-definite."