Differential geometry for physicists and mathematicians: moving frames and differential forms: from Euclid past Riemann

2015 ◽  
Vol 52 (07) ◽  
pp. 52-3715-52-3715
Keyword(s):  
Differential Geometry ◽  
Differential Forms ◽  
Moving Frames

Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

scholarly journals Moving Frames for Lie Pseudo–Groups

2008 ◽  
Vol 60 (6) ◽  
pp. 1336-1386 ◽  
Author(s):  
Peter J. Olver ◽  
Juha Pohjanpelto
Keyword(s):  
Recurrence Relations ◽  
Differential Forms ◽  
Group Actions ◽  
Differential Invariants ◽  
Moving Frames ◽  
Constructive Theory ◽  
Invariant Differential

AbstractWe propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. Themoving frame provides an effectivemeans for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

Differential forms on a Kähler manifold

1951 ◽  
Vol 47 (3) ◽  
pp. 504-517 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Complex Manifold ◽  
Differential Forms ◽  
Kähler Manifold ◽  
Systematic Account ◽  
General Plan ◽  
Kahler Manifold ◽  
Frequent Use ◽  
Compact Kähler Manifold

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

Differential forms in the model theory of differential fields

2003 ◽  
Vol 68 (3) ◽  
pp. 923-945 ◽  
Author(s):  
David Pierce
Keyword(s):  
Differential Geometry ◽  
Model Theory ◽  
Characteristic Zero ◽  
Differential Forms ◽  
Main Tool ◽  
Frobenius Theorem ◽  
Lie Bracket ◽  
Existentially Closed ◽  
Differential Fields

AbstractFields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

scholarly journals Differential forms with values in groups

1982 ◽  
Vol 25 (3) ◽  
pp. 357-386 ◽  
Author(s):  
Anders Kock
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Classical Theory ◽  
Lie Group ◽  
Differential Form ◽  
Differential Forms ◽  
Synthetic Differential Geometry ◽  
Exterior Differentiation ◽  
Group Object

In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.

scholarly journals Exterior Algebra with Differential Forms on Manifolds

2012 ◽  
Vol 60 (2) ◽  
pp. 247-252
Author(s):  
Md. Showkat Ali ◽  
K.M. Ahmed ◽  
M.R. Khan ◽  
Md. Mirazul Islam
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Differential Forms ◽  
Exterior Algebra ◽  
Linear Spaces ◽  
Exterior Differentiation

The concept of an exterior algebra was originally introduced by H. Grassman for the purpose of studying linear spaces. Subsequently Elie Cartan developed the theory of exterior differentiation and successfully applied it to the study of differential geometry [8], [9] or differential equations. More recently, exterior algebra has become powerful and irreplaceable tools in the study of differential manifolds with differential forms and we develop theorems on exterior algebra with examples.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11528 Dhaka Univ. J. Sci. 60(2): 247-252, 2012 (July)

INTEGRABLE SYSTEMS DETERMINED BY DIFFERENTIAL FORMS FOR MOVING FRAMES ON IMMERSED SUBMANIFOLDS

2008 ◽  
Vol 05 (07) ◽  
pp. 1041-1049 ◽  
Author(s):  
PAUL BRACKEN
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Integrable Systems ◽  
Differential Forms ◽  
Lax Pair ◽  
Dimensional Manifold ◽  
Moving Frames ◽  
Complete Set ◽  
The Given

It is shown how a system of differential forms can reproduce the complete set of differential equations generated by an SO(m) matrix Lax pair. By selecting the elements in the given matrices appropriately, examples of integrable nonlinear equations can be produced. The SO(3) case is discussed in detail, then extended to an m - 1 dimensional manifold immersed in Euclidean space.

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