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  • ...ar spaces and their mappings. The basic divisions of non-linear functional analysis are the following. ...ch, topological vector and certain more general spaces, including theorems on the local inversion of a differentiable mapping and the implicit-function t
    4 KB (490 words) - 17:11, 7 February 2011
  • [[Category:Global analysis]] Let $M$ and $N$ be two [[Differentiable manifold|$C^r$ manifolds]] and $f:M\to N$ a $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the
    1 KB (203 words) - 16:38, 17 November 2012
  • ...hod to construct global objects such as varieties, schemes, differentiable manifolds, vector bundles, sheaves $ \dots $ Consider, for example, the case of differentiable manifolds of dimension $ n $.
    4 KB (636 words) - 19:42, 5 June 2020
  • ...bundles. Attempts at a successive construction of topology on the basis of manifolds, mappings and differential forms date back to the end of 19th century (H. P .... Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and
    9 KB (1,298 words) - 14:59, 30 August 2014
  • ...TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Krzyzewski, "On analytic invariant measures for expanding mappings" ''Colloq. Math.'' , '' ...M. Shub, "Expanding maps" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 2
    2 KB (337 words) - 19:13, 9 October 2014
  • of linear transformations of the tangent spaces on a manifold $ M $ extended by linearity on $ T ^ {\mathbf C } M $)
    6 KB (880 words) - 16:10, 1 April 2020
  • Cartan's theorem on the highest weight vector. Let $ \mathfrak g $ on $ \mathfrak t $
    9 KB (1,307 words) - 20:04, 27 February 2021
  • and its inverse, defined on $f(X)$, is also denoted by $f^{-1}$. ...{\rm id}$. A notable example of involution is the [[complex conjugation]] on the [[Complex number|complex plane]]:
    10 KB (1,719 words) - 16:56, 30 November 2014
  • An [[Affine connection|affine connection]] on a Hermitian manifold $ M $ If the affine connection on $ M $
    3 KB (411 words) - 22:10, 5 June 2020
  • $#C+1 = 81 : ~/encyclopedia/old_files/data/I051/I.0501770 Integration on manifolds ...nalogues of differential calculus. There is also an "integral calculus on manifolds" . Let $ \Delta _ {n} = [ 0, 1] ^ {n} \subset \mathbf R ^ {n} $
    6 KB (827 words) - 22:13, 5 June 2020
  • A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form $ \omega $ the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space $ \mathbf C P ^ {n} $;
    4 KB (543 words) - 22:15, 5 June 2020
  • ...d the maps $f,g U \to \mathbb R$ are differentiable along a [[Vector field on a manifold|vector field]]. The rule becomes then ...b C$ and $f,g: U \to \mathbb C$ are differentiable in the sense of complex analysis (cf. [[Analytic function]]). Then the formula reads as \eqref{e:rule}.
    5 KB (757 words) - 10:34, 11 December 2013
  • A complex structure on a real vector space $ V $ is the structure of a complex vector space on $ V $
    3 KB (490 words) - 17:46, 4 June 2020
  • ...of tori and geodesic flows (cf. [[Geodesic flow|Geodesic flow]]) on closed manifolds of negative curvature are $Y$-systems. There are also other examples of a r The existence of a $Y$-system on a manifold imposes restrictions on the topological properties of the manifold. Little is known about this in g
    9 KB (1,321 words) - 07:59, 21 June 2014
  • A global version of the same statement is the following The global Theorem 2 holds also when $[0,T]$ is replaced by $[-T, 0]$ or $[-T,T]$, by
    5 KB (851 words) - 11:10, 30 November 2013
  • which is meromorphic on the entire manifold $ M $ In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.
    16 KB (2,223 words) - 17:31, 5 June 2020
  • ...ture of the boundary conditions or any supplementary conditions). Such a "global" character of variational calculus in the large proper is stressed by the ...f the number of closed geodesics (cf. [[Closed geodesic|Closed geodesic]]) on a closed Riemannian (and, more generally, a Finsler) manifold. This problem
    14 KB (2,052 words) - 08:27, 6 June 2020
  • ...of order $k$ reveals that $\varphi$ is completely determined by its values on the coordinate functions centred at this point $x _ { 0 }$. Thus, ...thbf{R} ^ { m }$ (cf. [[Weil algebra|Weil algebra]]) extends to all smooth manifolds and smooth mappings
    12 KB (1,895 words) - 17:46, 7 March 2021
  • ...rements, both of mathematics itself and of other sciences. In mathematics, manifolds arose first of all as sets of solutions of non-degenerate systems of equati ...utions of multi-dimensional variational problems (soap films), as integral manifolds of Pfaffian systems and of dynamical systems, as groups of geometric transf
    30 KB (4,462 words) - 07:59, 6 June 2020
  • cf. [[Tensor analysis|Tensor analysis]]) on a [[Manifold|manifold]] $ M $ ...iant under [[Parallel displacement(2)|parallel displacement]] along curves on $ M $.
    8 KB (1,160 words) - 08:05, 6 June 2020

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