By I. M. Gelfand, S. G. Gindikin, M. I. Graev
The miracle of indispensable geometry is that it's always attainable to recuperate a functionality on a manifold simply from the data of its integrals over yes submanifolds. The founding instance is the Radon remodel, brought before everything of the twentieth century. due to the fact that then, many different transforms have been discovered, and the overall idea used to be constructed. furthermore, many vital useful functions have been came across. the simplest recognized, yet under no circumstances the single one, being to clinical tomography.
This publication is a normal creation to quintessential geometry, the 1st from this standpoint for nearly 4 a long time. The authors, all top specialists within the box, characterize probably the most influential colleges in critical geometry. The e-book provides intimately simple examples of essential geometry difficulties, akin to the Radon remodel at the airplane and in area, the loo remodel, the Minkowski-Funk remodel, fundamental geometry at the hyperbolic aircraft and within the hyperbolic area, the horospherical rework and its relation to representations of $SL(2,\mathbb C)$, indispensable geometry on quadrics, and so on. The learn of those examples permits the authors to give an explanation for vital basic themes of vital geometry, equivalent to the Cavalieri stipulations, neighborhood and nonlocal inversion formulation, and overdetermined difficulties in vital geometry. some of the ends up in the booklet have been got by means of the authors during their career-long paintings in indispensable geometry.
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Selected Topics in Integral Geometry: 220
The miracle of fundamental geometry is that it is usually attainable to get well a functionality on a manifold simply from the data of its integrals over yes submanifolds. The founding instance is the Radon rework, brought initially of the twentieth century. due to the fact that then, many different transforms have been chanced on, and the final conception was once built.
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Extra resources for Selected Topics in Integral Geometry
Example text
Commutes with the operator d of exterior differentiation. We only need to note that the form w itself is closed as a form of maximal degree. To reconstruct the differential form w = I dx l /\ dx 2 from its Radon transform 'Rw = adp + bdcp, or. equivalently, to reconstruct the function I. p). We see that the Radon transform of 2-forms has trivial kernel. in contrast to the case of I-forms. We also note that the problem of reconstructing a differential 2-form w from its Radon transform 'R w is overdetermined because one must know only one coefficient of the form 'R w to find w completely.
7) w = h (X)dx2 /\ dx3 + h(x)dx3 /\ dx l + Ia(x)dx l /\ dx 2. X = (Xl. X2. X3), with coefficients in the Schwartz space S(R3). Integrating the form w over all p0ssible orientable planes in R3, we obtain a function on the manifold of planes. We refer to this function as the Radon translorm of wand denote it by'Rw. We present the expression for 'R. w in the coordinates on the manifold of planes in R3. 8) On the manifold of planes we take the coefficients al. 02. {3 of these equations for the local coordinates.
P) satisfy the conditions of the theorem. ) = -2 11" f+oc "'({, p)e -00 ip dp. I. RADON TRANSFORM 16 It follows from the homogeneity condition 1) that this relation is equivalent to the formula F(~~) 1 = -2 j+x 'P(~. p)ei>'p dp. -oc 1r Conditions 2) and 3) imply that the function F is infinitely differentiable at any point ~ ::F 0 and infinitely differentiable in any direction at ~ = 0: hence. it is infinitely differentiable at the point ~ = 0 by Cavalieri's conditions. Further, by conditions 2) and 3).