Spin(9)-invariant valuations
- The first aim of this thesis is to give a Hadwiger-type theorem for the exceptional Lie group Spin(9). The space of Spin(9)-invariant k-homogeneous valuations is studied through the construction of an exact sequence involving some spaces of differential forms. We present then a description of the spin representation using the properties of the 8-dimensional division algebra of the octonions. Using this description as well as representation-theoretic formulas, we can compute the dimensions of the spaces of differential forms appearing in the exact sequence. Hence we obtain the dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations for k=0,1,...,16. In the second part of this work, we construct one new element for a basis of one of these spaces. It is clear, that the k-th intrinsic volume is also Spin(9)-invariant. The last chapter of this work presents the construction of a new 2-homogeneous Spin(9)-invariant valuation. On a Riemannian manifold (M,g), we construct a valuation by integrating the curvature tensor over the disc bundle. We associate to this valuation on M a family of valuations on the tangent spaces. We show that these valuations are even and homogeneous of degree 2. Moreover, since the valuation on M is invariant under the action of the isometry group of M, the induced valuation on the tangent space in a point p in M is invariant under the action of the stabilisator of p for all p in M. In the special case where M is the octonionic projective plane, this construction yields an even, homogeneous of degree 2, Spin(9)-invariant valuation, whose Klain function is not constant, i.e. which is linearly independent of the second intrinsic volume.
Author: | Floriane Voide |
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URN: | urn:nbn:de:hebis:30:3-300700 |
Referee: | Andreas BernigORCiDGND, Gil Solanes |
Document Type: | Doctoral Thesis |
Language: | English |
Date of Publication (online): | 2013/06/11 |
Year of first Publication: | 2013 |
Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
Granting Institution: | Johann Wolfgang Goethe-Universität |
Date of final exam: | 2013/03/27 |
Release Date: | 2013/06/11 |
Tag: | octonions; spin group; valuation |
Page Number: | V, 55 |
HeBIS-PPN: | 322599261 |
Institutes: | Informatik und Mathematik |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Sammlungen: | Universitätspublikationen |
Licence (German): | Deutsches Urheberrecht |