Homotopy invariants for tangent vector fields on closed sets

2006 ◽  
Vol 65 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Aleksander Ćwiszewski ◽  
Wojciech Kryszewski
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Closed Sets ◽  
Homotopy Invariants

scholarly journals Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1547
Author(s):  
Stephen C. Anco ◽  
Bao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Evolution Equations ◽  
Applied Mathematics ◽  
Tangent Vector ◽  
Solution Space ◽  
Partial Differential ◽  
Geometrical Meaning ◽  
Geometrical Formulation

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

scholarly journals Tangent vectors to sets in the theory of geodesics

1987 ◽  
Vol 106 ◽  
pp. 29-47 ◽  
Author(s):  
Dumitru Motreanu
Keyword(s):  
Closed Subset ◽  
Tangent Vector ◽  
General Concept ◽  
Recent Book ◽  
Closed Sets ◽  
Banach Manifolds ◽  
Flow Invariance ◽  
Tangent Vectors

In the setting of Banach manifolds the notion of tangent vector to an arbitrary closed subset has been introduced in [11] by the author and N. H. Pavel, and it has been used in flow-invariance and optimization ([11], [12], [13]). For detailed informations on tangent vectors to closed sets (including historical comments) we refer to the recent book of N. H. Pavel [17].The aim of this paper is to apply this general concept of tangency in the study of geodesies. We are concerned with geodesies which have either the endpoints in given closed subsets or the same angle for a fixed closed subset. This approach allows one to extend important results due to K. Grove [4] and T. Kurogi ([6], [7]).

scholarly journals Inverse potential problems in divergence form for measures in the plane

2021 ◽  
Author(s):  
Laurent Baratchart ◽  
Douglas Hardin ◽  
Cristobal Villalobos-Guillén
Keyword(s):  
Vector Fields ◽  
Regularization Parameter ◽  
Tangent Vector ◽  
Thin Plates ◽  
Total Variation Norm ◽  
Infinite Dimensional ◽  
Jordan Curves ◽  
Norm Minimization ◽  
Potential Problems ◽  
Noise Limit

We study inverse potential problems with source term the divergence of some unknown (R 3 -valued) measure supported in a plane; e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization μ by penalizing the measure-theoretic total variation norm kμk T V , and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane that T V -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree- like. We note that such magnetizations can be recovered via T V -regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

scholarly journals Some lie algebras of vector fields and their derivations Case of partially classical type

1981 ◽  
Vol 82 ◽  
pp. 175-207 ◽  
Author(s):  
Yukihiro Kanie
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Tangent Vector ◽  
Classical Type ◽  
Foliated Manifold

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

scholarly journals Some relations between differential geometric invariants and topological invariants of submanifolds

1976 ◽  
Vol 60 ◽  
pp. 1-6 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Vector Fields ◽  
Tangent Vector ◽  
Second Fundamental Form ◽  
Dimensional Euclidean Space ◽  
Topological Invariants ◽  
Dimensional Manifold ◽  
Geometric Invariants ◽  
The Second Fundamental Form

Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by(1.1) ∇̃XY = ∇XY + h(X,Y).

scholarly journals Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 919
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Young Ho Kim
Keyword(s):  
Hyperbolic Space ◽  
Dual Space ◽  
Vector Fields ◽  
Tangent Vector ◽  
De Sitter Space ◽  
Normal Curve ◽  
De Sitter ◽  
Space Forms ◽  
Null Curve

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.

scholarly journals Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds

1999 ◽  
Vol 59 (2) ◽  
pp. 271-295
Author(s):  
Victor Brunsden
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Group Cohomology ◽  
Finitely Generated ◽  
Rigidity Result ◽  
Finitely Generated Group ◽  
Local Rigidity ◽  
Banach Manifolds ◽  
The Stability ◽  
Smooth Closed Manifold

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

Sign in / Sign up

Export Citation Format

Share Document