Geodesic
Maxwell's Equations, the Euler Index, and Morse Theory
Matematičeskie zametki, Tome 100 (2016) no. 3, pp. 331-343

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We show that the singularities of the Fresnel surface for Maxwell's equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.
Keywords: conical refraction, tensor, vector bundle, singularities.
Mots-clés : Fresnel surface, section 
@article{MZM_2016_100_3_a1,
     author = {C. Valero},
     title = {Maxwell's {Equations,} the {Euler} {Index,} and {Morse} {Theory}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {331--343},
     publisher = {mathdoc},
     volume = {100},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_3_a1/}
}
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C. Valero. Maxwell's Equations, the Euler Index, and Morse Theory. Matematičeskie zametki, Tome 100 (2016) no. 3, pp. 331-343. http://geodesic.mathdoc.fr/item/MZM_2016_100_3_a1/