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...tion <math>\nabla</math>''' is a [[Connection (vector bundle)|connection]] on a [[Hermitian vector bundle]] <math>E</math> over a smooth manifold <math>M <math>\langle \cdot, \cdot \rangle</math> on <math>E</math>, meaning that ...

{{Short description|Geometric structure on a smooth manifold}} ...'' is a certain kind of geometric structure on a [[smooth manifold]]. Such structures were introduced by [[Shigeo Sasaki]] in 1960. ...

...ture]] on [[orientable]] [[Riemannian manifold]]s. A metaplectic structure on a [[symplectic manifold]] allows one to define the [[symplectic spinor bund Symplectic spin structures have wide applications to [[mathematical physics]], in particular to [[quan ...

...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. ...

...>{{cite journal |author=Balázs Csikós and Márton Horváth |year=2011 |title=On the volume of the intersection of two geodesic balls |journal=Differential ...globally Osserman''' if the spectrum depends neither on <math>X</math> nor on <math>p</math>. All two-point [[homogeneous space]]s are globally Osserman, ...

...''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 ...inspired by the analytic continuation property of analytic diffeomorphisms on an [[analytic manifold]]). ...

...ion of the dynamical system.<ref>Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977 Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an [[invariant subspace]] about an e ...

...ath>\phi : C \to C' \subset \partial \Delta_{n+1}</math>, the Pachner move on ''N'' associated to ''C'' is the triangulated manifold <math>(N \setminus C *{{citation|first=Udo|last=Pachner|title=P.L. homeomorphic manifolds are equivalent by elementary shellings|journal=[[European Journal of Combin ...

for every vector field <math>X</math> on <math>M</math>. ...most complex structure on the six-sphere is not induced by a complex atlas on <math>S^6</math>. ...

...on structure]]s, and having several applications to mechanics. It is based on the notion of the [[Dirac bracket]] constraint introduced by [[Paul Dirac]] == Linear Dirac structures == ...

...onic projective space]] which should clearly be considered as quaternionic manifolds. ...te journal |first=Vivian Yoh |last=Kraines |title=Topology of quaternionic manifolds |journal= Transactions of the American Mathematical Society |volume=122 |is ...

{{Short description|In mathematics, on the region between two well-behaved spheres}} ...\mathbb{D}^{n}</math> into <math>\mathbb{R}^{n}</math>, where then smooth structures can be pulled back along the immersion and be lifted to covers. ...

...ted to professor G. B. Rizza, who is the originator of the notion of Rizza manifolds.''"</ref> is an [[almost complex manifold]] also supporting a [[Finsler man ...g this structure "''Rizza structure''", and manifolds carrying it "''Rizza manifolds''".<ref name="Ich88"/> ...

...-540-05467-2 |mr=0285027 |authorlink=André Haefliger}}</ref> Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely de A codimension-''<math>q</math>'' '''Haefliger structure''' on a topological space ''<math>X</math>'' consists of the following data: ...

...r, J. W.|title=On the Construction and Classification of Exotic Structures on the 7-Sphere|postscript=; Bachelor's thesis|year=2017|publisher=Utrecht Uni ...rn |title=Examples of singular normal complex spaces which are topological manifolds |journal=[[Proceedings of the National Academy of Sciences of the United St ...

...perturbations. Thus, in problems involving perturbation theory, invariant manifolds exist with certain hyperbolicity properties, which can in turn be used to o ...''E''^''u'' must be an expansion, and must be relatively neutral on <math> T\Lambda </math>. Thus, there exist constants <math> 0 < \lambda < \ ...

...|doi=10.1090/gsm/020|mr=1707327}}</ref><ref>A.Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. {{isbn|0-8218-3749-4}}</ref> ...=Selman Akbulut|first2=Kouichi| last2=Yasui| title=Corks, plugs and exotic structures|url=http://gokovagt.org/journal/2008/jggt08-akbulutyasui.pdf| journal= ...

..., results concerning the scalar curvature, and results on spinors and spin structures. ...''M'' and a [[spinor bundle]] ''S'', the Lichnerowicz formula states that on a [[section (fiber bundle)|section]] &psi; of ''S'', ...

...ns, Kervaire–Milnor groups furthermore give the possible smooth structures on spheres, hence [[Exotic sphere|exotic spheres]]. They are named after the F ...ossible with [[Homotopy sphere|homotopy spheres]], which are closed smooth manifolds with the same homotopy type as a sphere, with restriction to h-cobordism cl ...

...''Introduction to Smooth Manifolds'', Springer-Verlag, 2003.</ref> A note on notation: in this article, we denote projection maps by their domains, e.g. ...equations, i.e., [[spray (mathematics)|(semi)spray structures]] on smooth manifolds, and it is not to be confused with the [[jet bundle|second order jet bundle ...