denotes the [[Lie algebra|Lie algebra]] of vector fields on $ M $),
...Riemannian curvature tensor, the Ricci operator, and the scalar curvature on $ M $,
15 KB (2,270 words) - 08:28, 26 March 2023
...ww.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634601.png" /> (or
on a Riemann surface <img align="absmiddle" border="0" src="https://www.encycl
...aofmath.org/legacyimages/m/m063/m063460/m06346020.png" />. It follows that
on a non-compact Riemann surface <img align="absmiddle" border="0" src="https:
44 KB (5,974 words) - 22:47, 29 November 2014
A Hermitian metric on a complex vector space $ V $
is a positive-definite [[Hermitian form|Hermitian form]] on $ V $.
4 KB (681 words) - 03:41, 21 March 2022
A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphic [[Vector bundle|vector
if on $ E $
7 KB (1,004 words) - 08:07, 6 June 2020
which is called a Morse–Smale diffeomorphism in this case) on a compact (usually closed) differentiable $ m $-
...ed system). This guarantees the existence of stable and unstable invariant manifolds $ W ^ {s} $
10 KB (1,474 words) - 20:08, 12 January 2024
Mathematical background on dynamical systems can be found in [[#References|[a3]]], [[#References|[a7]]
...has a strictly positive part. In the remaining cases the stability depends on the non-linear terms in the Taylor expansion of $G$ (cf. also [[Stability t
13 KB (1,928 words) - 17:00, 1 July 2020
...rties and applications. Moreover, these integral formulas typically depend on the domain under consideration, and they intimately reflect complex-analyti
...ves iteration of the one-variable formula on product domains. For example, on a poly-disc $P = \{ ( z _ { 1 } , \dots , z _ { n } ) : | z _ { j } - a _ {
15 KB (2,167 words) - 16:10, 11 February 2024
...85 : ~/encyclopedia/old_files/data/D032/D.0302180 Differential geometry of manifolds
...simal structures (cf. [[Infinitesimal structure|Infinitesimal structure]]) on a manifold and their connection with the structure of the manifold and its
30 KB (4,323 words) - 19:35, 5 June 2020
A closed smooth curve on a [[Riemannian manifold|Riemannian manifold]] $ M $
not constant on any subinterval, is a closed geodesic if, for sufficiently small $ \epsil
13 KB (1,911 words) - 17:44, 4 June 2020
of functions bounded on the interval $ ( 0, 1) $
...orems of the second type, estimates of the norms of traces of functions on manifolds of smaller dimension in terms of their weighted norms are given.
9 KB (1,435 words) - 08:13, 13 January 2024
...] - \infty , 0 ] )$ is bounded and the $f_j$ do not vanish simultaneously on $\partial K$. Letting $f = ( f _ { 1 } , \dots , f _ { n } )$, Kronecker sh
...d manifolds of the same finite dimension. He used it to prove the theorems on invariance of dimension and invariance of domain (cf. also [[Brouwer theore
12 KB (1,815 words) - 17:42, 1 July 2020
on $ X $;
...ing analytic set in a local model (cf. [[Analytic set|Analytic set]]). The global dimension is defined by the formula:
22 KB (3,277 words) - 01:53, 19 January 2022
...s of topological spaces and extremal properties of functions (functionals) on them. Morse theory is a branch of [[Variational calculus in the large|varia
on a smooth manifold $ M $(
21 KB (3,095 words) - 08:01, 6 June 2020
...n-diffusion systems [[#References|[a19]]], and fluctuations in fluid films on inclines [[#References|[a30]]]. Indeed, (a1) generically describes the dyna
...e that as suggested by the presence of chaotic solutions and by a Painlevé analysis [[#References|[a7]]] (cf. also [[Painlevé test|Painlevé test]]), the Kura
21 KB (3,050 words) - 17:43, 1 July 2020
with coefficients depending on variables $ x $,
The type of a Monge–Ampère equation depends on the sign of the expression
17 KB (2,601 words) - 08:01, 6 June 2020
'''On a ''general'' complex quintic threefold, there are only finitely many smoot
.../math> is the projectivization of the kernel of the natural restriction of global sections
14 KB (2,200 words) - 18:27, 23 October 2017
...the general framework of the theory of connections. As a device of tensor analysis, covariant differentiation is widely used in theoretical physics, particula
Let an affine connection be given on an $ n $-
15 KB (2,257 words) - 17:31, 5 June 2020
...ransformation) to general position" , the precise meaning of which depends on the context. Usually the set $\mathfrak{O}$ of all objects considered has a
...eft({\lambda_1, \dots, \lambda_n}\right)$, depending sufficiently smoothly on $n$ (scalar) parameters, and when all possible such families form a Baire s
12 KB (1,758 words) - 00:29, 13 January 2017
...analysis that a bounded and [[Holomorphic function|holomorphic function]] on the whole plane (cf. also [[Entire function|entire function]]) must be a co
...ypercone in $\mathbb R^n$ must be planar for $n\leq 7$. His proof is based on an inequality for the norm of the second fundamental form of minimal surfac
10 KB (1,518 words) - 21:22, 14 January 2021
...uctive representations of commutative algebras characterizing differential manifolds in non-commutative algebras. The most remarkable examples are geometric qua
..., as the quotient with respect to gauge groups produces finite-dimensional manifolds [[#References|[a2]]], [[#References|[a34]]], [[#References|[a35]]], [[#Refe
16 KB (2,356 words) - 19:53, 13 January 2018