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Wave in Vacuum: Known Geometrical Results

Let’s take a purely geometrical example: assume that the wave surface is in vacuum. Assume further that the first fundamental form of the wave surface is

while the second fundamental form is

where (,) represents the usual dot product of vectors. Now, assume that the second fundamental form of the wave surface varies, i.e. the wave changes the direction of propagation, and as in vacuum there is nothing to determine the variation of the wave surface, for instance a surface of separation between two media, this variation can be only “assisted” by the first fundamental form of the wave surface itself. As a matter of fact, one can say that the wave surface is the only surface of separation in vacuum: in front of it there is no light, all the light is coming behind it. By the formal considerations above, the pure variation due to curvature variation takes place along the solutions of the differential equation (3) which, in this case is given by

Secondly, assume that the “assisting” matrix of new parameters from equation (6) is given by the first fundamental form of the wave surface (it can be given by the second fundamental form just as well) i.e.

In this case we can write the condition of integrability of (17) as

The matrix (12) is therefore given by

with the differential forms given by equation (13) as

The trace of matrix (20) reproduces the known result about the mean curvature

On the other hand, the determinant of this matrix gives the absolute curvature:

Consequently, when taking everything as variations rather than absolute values, this approach reproduces the classical results: equation (22) represents the mean curvature, while equation (23) represents the Gaussian curvature. This particular case shows that by this theory we can recover the classical results from the Differential Geometry. But the theory contains more than these classical results in two essential respects, and this is what interests us most.

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