$#C+1 = 113 : ~/encyclopedia/old_files/data/K055/K.0505350 Kervaire invariant
An invariant of an almost-parallelizable smooth manifold $ M $
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An invariant differential operator is a [[Differential operator|differential operator]]
...metric tensor. In the theory of Lie groups, the so-called left- and right-invariant operators under the corresponding shifts on the group are of great importan
3 KB (392 words) - 21:35, 24 July 2012
...ds, that is, sets $S_t$ that are tori for any fixed $t\in\mathbf R$. These manifolds are widely encountered in systems of type \eqref{*} describing oscillatory
...op">[6]</TD> <TD valign="top"> Yu.A. Mitropol'skii, O.B. Lykova, "Integral manifolds in non-linear mechanics" , Moscow (1973) (In Russian)</TD></TR></table>
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...contact metric structure is called a Sasakian manifold. To study Sasakian manifolds one often uses the following characterization (cf. [[#References|[a4]]]): A
General references for Sasakian manifolds are [[#References|[a2]]], [[#References|[a3]]], [[#References|[a6]]].
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An invariant of oriented links (cf. also [[Link]]).
...lynomial can be generalized to homotopy skein modules of three-dimensional manifolds (cf. also [[Skein module]]).
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...hese spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation $Ax=b$.
...l differential operators acting on sections of vector bundles over compact manifolds.
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is invariant under $ J $,
is called a totally real (anti-invariant) submanifold if $ J ( T _ {x} N ) \subset T _ {x} N ^ \perp $
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...possibility of algorithmically recognizing homeomorphy of four-dimensional manifolds. The exceptional status of dimension 4 is well-illustrated by the following
There exist four-dimensional manifolds that do not admit piecewise-linear structures. If such a structure exists a
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...groups]]). It encompasses the theory of quantum invariants of knots and 3-manifolds, [[Algebraic topology based on knots|algebraic topology based on knots]], $
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...manifold with an almost-complex structure is a Lie group. For non-compact manifolds this statement is, in general, not true.
This represents an invariant complex structure in the tangent space $ T _ {p} M $
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$#C+1 = 51 : ~/encyclopedia/old_files/data/P073/P.0703770 Pontryagin invariant
An invariant of framed constructions of surfaces with a given framing. Let $ ( M ^ {2
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...\{~,~\})$ is called a '''Poisson manifold'''. A smooth map between Poisson manifolds $\phi:(M,\{~,~\}_M)\to (N,\{~,~\}_N)$ such that the induced pullback map $\
== Examples of Poisson manifolds ==
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...ven simpler (thus, an expanding mapping of class $C^2$ always has a finite invariant measure, defined in terms of the local coordinates as a positive density).
...<TD valign="top">[5]</TD> <TD valign="top"> K. Krzyzewski, "On analytic invariant measures for expanding mappings" ''Colloq. Math.'' , '''46''' : 1 (1982)
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therefore a spray describes (and moreover in an invariant manner, that is, independent of the coordinate system) a system of such equ
...efinition of a spray in invariant terms, which is suitable also for Banach manifolds (see [[#References|[1]]]).
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...modifications). The depth of the centre is not large for flows on compact manifolds of dimension $\leq2$ (see [[#References|[5]]], [[#References|[6]]]) or for
...which $S_tx\in U$ tends to 1 as $T\to\infty$. However, the smallest closed invariant set having this property (the minimal centre of attraction, see [[#Referenc
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$\operatorname{CS}$ is invariant under gauge transformations, i.e. automorphisms of the $G$-bundle, and henc
...s|[a6]]] defined invariants for homology $3$-spheres related to the Casson invariant (see [[#References|[a7]]]).
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''$ \eta $-invariant''
...al paper [[#References|[a1]]] as a correction term for an index theorem on manifolds with boundary (cf. also [[Index formulas|Index formulas]]). For example, in
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have been intensively studied on Kähler manifolds; see, e.g., [[#References|[a3]]], [[#References|[a16]]], [[#References|[a17
...ified with the fourth-order Chern–Moser tensor [[#References|[a4]]] for CR-manifolds.)
15 KB (2,270 words) - 08:28, 26 March 2023
...e same example was the first example of homeomorphic but not diffeomorphic manifolds.
be the set of classes of $ h $-cobordant $ n $-dimensional smooth manifolds which are homotopically equivalent to $ S ^ {n} $.
10 KB (1,339 words) - 08:44, 18 March 2023
...the first observations of asymmetric manifolds, which were particular lens manifolds (cf. [[Lens space|Lens space]]).
...TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin, "Smooth manifolds and their applications in homotopy theory" ''Transl. Amer. Math. Soc.'' ,
3 KB (548 words) - 20:58, 1 January 2019