# Search results

• ...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

View (previous 20 | next 20) (20 | 50 | 100 | 250 | 500)