Project Materials

Differential Forms and Applications

TABLE OF CONTENTS

1 MANIFOLDS AND FORMS 2
1.1 Sub-manifolds of Rn without boundary . . . . . . . . . . . . . 2
1.2 Notions of forms and elds . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 forms and vector elds on Rn . . . . . . . . . . . . . . 10
1.2.3 Integration over cubes and chains . . . . . . . . . . . . 15
1.3 Classical theorems of Green and Stokes . . . . . . . . . . . . . 18
1.3.1 Orientable Manifolds . . . . . . . . . . . . . . . . . . . 20
1.3.2 Riemannian Manifolds . . . . . . . . . . . . . . . . . . 27
2 EXAMPLES OF DIFFERENTIAL FORMS ON RIEMAN-
NIAN MANIFOLDS 28
2.1 Winding form and volume element associated to ellipsoids in
R2 and in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Differential forms on the 1-dimensional ellipsoid . . . . 28
2.1.2 Differential forms on the 2-dimensional ellipsoid . . . . 33
2.2 Other quantities associated to R3 ellipsoid derived from Riemannian
structure, geodesics of R3 ellipsoid . . . . . . . . . . 39
2.2.1 The shape operator . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Geodesics of the 2-dimensional ellipsoid . . . . . . . . . 42
2.3 Manifolds in higher dimensions: volume element, geodesics . . 48
2.3.1 Higher dimensional volume forms . . . . . . . . . . . . 48
2.3.2 Higher dimensional geodesics . . . . . . . . . . . . . . 52
Bibliography 58

CHAPTER ONE

MANIFOLDS AND FORMS

1.1 Sub-manifolds of Rn without boundary

Definition 1.1.1.

A subset M of Rn is called a k-dimensional sub-manifold without boundary if for each point p 2 M, there exist U; V open in Rn with p 2 U as well as a dimorphism : U ! V such that (U M) is contained in the subspace
Rk Rn. In other words, (U M) = V (Rk f0gn􀀀k) = fy 2 V : yk+1 = = yn = 0g:

The pair (U; ) where U = U M is called a local chart around p and a family of local charts covering all points of M is called an atlas on M. Thus if fUi; igi2IN is an atlas on M, then M = [i2I
Ui:

Remark: Because the dimension of M is k, we say that M has a local Rk property and use this property to create Parametrization for the manifold, which are basically differentiable functions mapping from a subset of Rk onto M. Parametrization are needed for computational and analytical purposes as we will see in chapter 2 on examples of differential forms on Riemannian manifolds.

If (U; ) is a local chart with p 2 U, we can identify p and the vector (p) 2 Rn. The coordinates of (p) in Rn are called the local coordinates 2 of p in the local chart (U; ): For any two charts (Ui; i) and (Uj ; j) such that Ui Uj is non-empty, we can dene the map,
i j
􀀀1 : j(Ui Uj) ! i(Ui Uj)
which is called a chart transition from one chart to another. The sets j(Ui Uj) and i(Ui Uj) are open sets of the coordinate space Rk and and the transition function i j 􀀀1 is a dimorphism. Alternatively, we may dene a sub-manifold of Rn without boundary as follows.