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In their "natural" state, they are – a familiar picture indeed – some kind of rippling liquid collective life form, a manifold united into an ocean of calm waves, a biological orbifold.
The equation should be understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance, some kind of constant curvature.
A space can be endowed with various kinds of geometric structures: a real differentiable structure describes a manifold, a complex structure describes a complex manifold, and a Riemannian structure describes a manifold together with a metric.
A deckhand tied the ship off while another hooked its hoses to a manifold.
It's an unusual film in the Ozu canon and a manifold delight on its own.
The burden of proof is a manifold and somewhat ambiguous concept in the law of evidence.
A Nash manifold denotes a real manifold furnished with algebraic structure, following a theorem of Nash that a compact differentiable manifold can be imbedded in a Euclidean space so that the image is precisely such a manifold.
A manifold which is an affine manifold is homogeneous in this sense.
If formulated as above, a spacetime is a manifold of events with certain fields defined on the manifold.
A manifold T ∈ R is a one-dimensional smooth manifold called a time manifold.
A metric \(g\) on a manifold determines a first-order conformal structure on the manifold, namely, an equivalence class of conformally related metrics A metric \(g\) also uniquely determines a symmetric linear connection \ \Gamma\) on the manifold.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com