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{{Short description|Type of Riemannian metric}} ...ric''' is a natural choice of Riemannian metric on the tangent bundle of a Riemannian manifold. ...

...mov]].<ref>{{cite journal |last=Gromov |first=M. |title=Filling Riemannian manifolds |journal=J. Diff. Geom. |volume=18 |date=1983 |pages=1–147 |citeseerx=10.1. *All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere ''S²''. ...

...fold''' is an [[almost-contact manifold]] endowed with a certain kind of [[Riemannian metric]]. They are named after the Japanese mathematician Katsuei Kenmotsu. ...hi, \xi, \eta)</math> be an [[almost-contact manifold]]. One says that a [[Riemannian metric]] <math>g</math> on <math>M</math> is adapted to the almost-contact ...

{{Short description|The isometry group of a Riemannian manifold is a Lie group}} ...unction|smooth]] [[Isometry (Riemannian geometry)|isometry]] of Riemannian manifolds. A simpler proof was subsequently given by [[Richard Palais]] in 1957. The ...

...connection coincides with the [[Levi-Civita connection]] of the associated Riemannian metric. * Shiing-Shen Chern, ''Complex Manifolds Without Potential Theory''. ...

...to [[Cartan–Hadamard conjecture]], it should hold in all simply connected manifolds of nonpositive curvature. [[Category:Riemannian geometry]] ...

...ly, non Kählerian nearly Kähler manifolds are called "strict nearly Kähler manifolds". Nearly Kähler manifolds, also known as almost Tachibana manifolds, were studied by Shun-ichi Tachibana in 1959<ref> ...

...a''') is a fundamental equation in the analysis of [[spinor]]s on [[pseudo-Riemannian manifold]]s. In dimension 4, it forms a piece of [[Seiberg–Witten gauge th Given a [[spin structure]] on a pseudo-Riemannian manifold ''M'' and a [[spinor bundle]] ''S'', the Lichnerowicz formula stat ...

{{short description|Type of Riemannian manifold with constant Jacobi operator spectrum}} ...rticularly in [[differential geometry]], an '''Osserman manifold''' is a [[Riemannian manifold]] in which the [[characteristic polynomial]] of the [[Jacobi opera ...

...ries.'' In: Springer LNM, 1620 (1996), pp.&nbsp;120–348.</ref> is a flat [[Riemannian manifold]] with a certain compatible multiplicative structure on the [[tang ...the discussion here to smooth (real) manifolds. A restriction to complex manifolds is also possible. ...

...r the n-dimensional cube <math>[0,1]^n</math> with a [[Riemannian geometry|Riemannian]] metric <math>g</math>. Let ...equality can be generalized in the following way. Given an n-dimensional [[Riemannian manifold]] M with connected boundary and a smooth map <math>f: M \rightarro ...

...p.com/product/9780198503620.do|title=Harmonic Morphisms Between Riemannian Manifolds|work=Oxford University Press}}</ref> ...mplex-valued harmonic morphisms <math>\phi:(M,g)\to\mathbb C</math> from [[Riemannian manifold]]s generalise [[holomorphic function]]s <math>f:(M,g,J)\to\mathbb ...

...anifold. Broadly speaking, [[projective geometry]] refers to the study of manifolds with this kind of connection. ...manifolds with a conformal equivalence class of Riemannian metrics, i.e., manifolds modeled on the conformal sphere. Here the associated Cartan connection is ...

...s there are various types of manifolds, there are various types of maps of manifolds. ...DIFF]], the category of [[piecewise]]-smooth maps between piecewise-smooth manifolds. ...

...''X'')-structure are always [[manifold]]s and are called '''(''G'', ''X'')-manifolds'''. This notion is often used with ''G'' being a [[Lie group]] and ''X'' a === Riemannian examples === ...

| known_for = [[Riemannian hyperbolization]] ...onship between exotic structures and negative or non-positive curvature on manifolds. ...

...les''', especially in connection with the [[filling area conjecture]] in [[Riemannian geometry]],{{r|katz}} but this term has also been used for other concepts.{ ...ording to which the unit hemisphere is the minimum-area surface having the Riemannian circle as its boundary.{{r|gromov}} ...

...of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds | last1 = Chanu | first1= Claudia | last2 = Rastelli|first2 = Giovanni|jour An [[orthogonal]] '''web''' on a [[Riemannian manifold]] ''(M,g)'' is a set <math>\mathcal S = (\mathcal S^1,\dots,\mathc ...

...eorem''' is a theorem of [[Riemannian geometry]], according to which the [[Riemannian metric]] is locally determined by the [[Riemann curvature tensor]], or in o ...78|issn=0003-486X}}</ref> This was further generalized by Hicks to general manifolds with [[Affine connection|affine connections]] in their [[Tangent bundle|tan ...

...also symmetric. Such tensors arise naturally in the study of [[Riemannian manifolds]] with [[harmonic]] [[curvature]] or harmonic [[Weyl tensor]]. In fact, exi Let <math>(M,g)</math> be a n-dimensional Riemannian manifold for <math>n \geq 3</math>, let <math>T</math> be a symmetric 2-[[t ...