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Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space

### Abstract

In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green’s operator, an explicit formula of Biot–Savart type for the magnetic field, and a corresponding Ampere’s law contained in Maxwell’sequations. We then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological because their integrands are invariant under orientation-preserving isometries of the ambient space. In the second paper, we obtain integral formulas for twisting, writhing, and helicity and prove the theorem in the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of most of these results and a hint of their proofs can be found in e-print arXiv:math.GT/0406276, while an expanded version of this paper, containing many details which are here left to the reader, can be found at e-print arXiv:math.GT/0510388.

© 2008 American Institute of Physics

Received 26 March 2007
Accepted 28 November 2007
Published online 08 February 2008

Acknowledgments:
We are grateful to Jason Parsley for his help throughout the preparation of this paper and refer the reader to his Ph.D. thesis^{17} for a study of electrodynamics, linking, writhing, and helicity on bounded subdomains of the 3-sphere. We also thank Shea Vela-Vick for his critical reading and analysis of this manuscript over the past year, Benjamin Schak and Clayton Shonkwiler for their more recent very thorough reading, and Jozef Dodziuk and Charles Epstein for a number of helpful consultations.

Article outline:

I. ORGANIZATION
II. STATEMENTS OF RESULTS
A. Linking integrals in , , and
B. The route to the Gauss linking integral
C. Magnetic fields in , , and
D. Left-invariant vector fields on
E. Scalar-valued Green’s operators in , , and
F. Vector-valued Green’s operators in , and
G. Classical electrodynamics and Maxwell’sequations on and
III. PROOFS ON IN LEFT-TRANSLATION FORMAT
A. Proof scheme for Theorem 3, formula (14)
B. The calculus of vector convolutions
C. Proof of Theorem 3, formula (14)
D. Proof of Theorem 2, formula (7)
IV. PROOFS ON IN PARALLEL TRANSPORT FORMAT
A. Parallel transport in
B. Geodesics in
C. The vector triple product in
D. Explicit formula for parallel transport in
E. The method of moving frames in
F. Curls and divergences
G. Statement of the Key Lemma
H. The first term on the left hand side of the Key Lemma
I. The second term on the left hand side of the Key Lemma
J. The right hand side of the Key Lemma
K. Maxwell’sequation on
L. Proof of Theorem 2, formula (8)
M. The Green’s operator on in parallel transport format
N. Plan of the proof
O. A useful formula
P. The curl of
Q. The divergence of
R. Step 2: Finding and using the kernel function
S. Finding an explicit formula for
T. Explicit formula for
V. PROOFS IN IN PARALLEL TRANSPORT FORMAT
A. The hyperboloid model of hyperbolic 3-space
B. Isometries of
C. Geodesics in
D. The vector triple product in
E. Explicit formula for parallel transport in
F. The method of moving frames in
G. Curls and divergences
H. Statement of the Key Lemma
I. Maxwell’sequation in
J. Proof of Theorem 2, formula (9)
K. The Green’s operator on in parallel transport format
VI. ELECTRODYNAMICS AND THE GAUSS LINKING INTEGRAL ON AND
A. Proof of Theorem 4
B. Proof scheme for Theorem 1
C. Proof of Theorem 1, formula (1)
D. Proof of Theorem 1, formulas (2) and (3)

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