Affinity connection

Parallel movement along a curve.

Affine connectivity is a linear connection on the tangent bundle of a manifold. Coordinate expressions of affine connectivity are Christoffel symbols .

On a smooth manifold, each point has its tangent space. Affine connection allows us to consider tangent spaces along one curve as belonging to one space; this identification is called parallel translation . Due to this, for example, differentiation operations of vector fields can be defined.

Content

Affine connectivity and tensor calculus

For more information see the article Covariant derivative .

In three-dimensional Euclidean space, the operation of differentiating vector fields is defined. When determining the derivative of a vector field on a manifold by such a formula, the obtained value is not a vector (tensor) field. That is, when the coordinates are replaced, it is not transformed according to the tensor law. In order for the result of differentiation to be a tensor, additional correction terms are introduced. These terms are known as Christoffel symbols .

Definition

Let M be a smooth manifold and $C^{\infty }(M,TM)$ ${\ displaystyle C ^ {\ infty} (M, TM)}$ denotes the space of vector fields on M. Then the affine connection on M is a bilinear map

{\begin{matrix}C^{\infty }(M,TM)\times C^{\infty }(M,TM)&\rightarrow &C^{\infty }(M,TM)\\(X,Y)&\mapsto &\nabla _{X}Y,\end{matrix}}

{\ displaystyle {\ begin {matrix} C ^ {\ infty} (M, TM) \ times C ^ {\ infty} (M, TM) & \ rightarrow & C ^ {\ infty} (M, TM) \\ ( X, Y) & \ mapsto & \ nabla _ {X} Y, \ end {matrix}}}

such that for any smooth function f ∈ C ^∞ ( M , R ) and any vector fields X , Y on M :

$\nabla _{fX}Y=f\nabla _{X}Y$ ${\ displaystyle \ nabla _ {fX} Y = f \ nabla _ {X} Y}$ , i.e, $\nabla$ ${\ displaystyle \ nabla}$ linearly in the first argument;
$\nabla _{X}(fY)=\mathrm {d} f(X)Y+f\nabla _{X}Y$ ${\ displaystyle \ nabla _ {X} (fY) = \ mathrm {d} f (X) Y + f \ nabla _ {X} Y}$ , i.e $\nabla$ ${\ displaystyle \ nabla}$ satisfies the Leibniz rule for the second variable.

Related Definitions

Torsion of affine connection $\nabla$ ${\ displaystyle \ nabla}$ called expression

T^{\nabla }(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y],

{\ displaystyle T ^ {\ nabla} (X, Y) = \ nabla _ {X} Y- \ nabla _ {Y} X- [X, Y],}

Where

[{*},{*}]

{\ displaystyle [{*}, {*}]}

means the Lie bracket of vector fields.

Zero- torsion affine connection on a Riemannian manifold with respect to which the metric tensor is covariantly constant $(\nabla g=0)$ ${\ displaystyle (\ nabla g = 0)}$ is called Levi-Civita connectivity .
The curvature of affine connectivity $\nabla$ ${\ displaystyle \ nabla}$ (or Riemannian curvature) is called the tensor
$R^{\nabla }(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.$ ${\ displaystyle R ^ {\ nabla} (X, Y) Z = \ nabla _ {X} \ nabla _ {Y} Z- \ nabla _ {Y} \ nabla _ {X} Z- \ nabla _ {[X , Y]} Z.}$
An affine connection with zero curvature is called Euclidean .

Literature

Original works

Christoffel, Elwin Bruno (1869), Über die Transformation der homogenen Differentialausdrücke zweiten Grades, J. Für die Reine und Angew. Math. T. 70: 46–70
Levi-Civita, Tullio (1917), " Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura Riemanniana ", Rend. Circ. Mat. Palermo T. 42: 173–205 , DOI 10.1007 / bf03014898
Cartan, Élie (1923), " Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) ", Annales Scientifiques de l'École Normale Supérieure T. 40: 325-412 , < http: // www .numdam.org / item? id = ASENS_1923_3_40__325_0 >
Cartan, Élie (1924), " Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite) ", Annales Scientifiques de l'École Normale Supérieure T. 41: 1–25 , < http: //www.numdam.org/item?id=ASENS_1924_3_41__1_0 >

In this work, the approach to the study of affine connectivity is motivated by the study of the theory of relativity. It includes a detailed discussion of reference frames , and how connectivity reflects the physical concept of moving along a world line .

Cartan, Élie (1926), " Espaces à connexion affine, projective et conforme ", Acta Math. T. 48: 1–42 , DOI 10.1007 / BF02629755

In this work, a more mathematical approach to the study of affine connectivity is used.

Cartan, Élie (1951), with appendices by Robert Hermann, ed., Geometry of Riemannian Spaces (translation by James Glazebrook of Leçons sur la géométrie des espaces de Riemann , 2nd ed.), Math Sci Press, Massachusetts, 1983, ISBN 978 -0-915692-34-7 , < https://books.google.com/?id=-YvvVfQ7xz4C&pg=PP1 > .

Affine connection is considered from the point of view of Riemannian geometry . An appendix written by Robert Herman discusses motivation from the point of view of surface theory, as well as the concept of affine connection in the modern sense and the main properties of the covariant derivative .

Weyl, Hermann (1918), Raum, Zeit, Materie (5 editions to 1922, with notes by Jürgen Ehlers (1980), translated 4th edition Space, Time, Matter by Henry Brose, 1922 (Methuen, reprinted 1952 by Dover) ed. ), Springer, Berlin, ISBN 0-486-60267-2

Contemporary Literature

Rashevsky P.K. Rimanova geometry and tensor analysis. - Any edition.
Kobayashi Sh. , Nomizu K. Fundamentals of differential geometry. - Novokuznetsk: Novokuznetsk Institute of Physics and Mathematics. - T. 1. - 344 p. - ISBN 5-80323-180-0 .
Dubrovin B.A., Novikov S.P., Fomenko A.T. Modern geometry. Methods and applications. - M.: Science, 1979.
M. Postnikov. Smooth manifolds (Lectures on geometry. Semester III) .