riemannian

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see also: Riemannian

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Adjective

riemannian (not comparable)

1. Alternative letter-case form of Riemannian#English|Riemannian

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Riemannian

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Adjective

riemannian (not comparable)

1. (mathematics) Of or relating to the work, or theory developed from the work, of German mathematician Bernhard Riemann, especially to Riemannian manifolds and Riemannian geometry.
   • 2003, Maung Min-Oo, The Dirac Operator in Geometry and Physics, Steen Markvorsen, Maung Min-Oo (editors), Global Riemannian Geometry: Curvature and Topology, Springer, page 62 ↗,

     Similarly, \hat A(M) is represented by the closed differential form

     \hat A(M) = \sqrt{\operatorname{det}}\left (\frac{R/2}{\sinh(R/2)} \right )

     where R is the Riemannian curvature of the metric g, regarded as an \textstyle\operatorname{End}(TM)-valued two form, and \sqrt{\operatorname{det}} is the Pfaffian, which is an invariant polynomial defined on the Lie algebra of skew symmetric matrices in even dimensions.

   • 2010, Charles P. Boyer, Krzysztof Galicki, Chapter 3: Sasakian geometry, holonomy, and supersymmetry, Vicente Cortés (editor), Handbook of Pseudo-Riemannian Geometry and Supersymmetry, European Mathematical Society, page 41 ↗,

     As the preferred metrics applied to symplectic forms are Kähler metrics one could ask for the Riemannian structure which would make the cone with the metric \overline g = dt^2 + t^2g together with the symplectic form \omega into a Kähler manifold. Then \overline g and \omega define a complex structure \overline \Phi. Alternatively, one could ask for a Riemannian metric g on M which would define a Kähler metric h on \mathcal Z via a Riemannian submersion.

   • 2012, Yves Carriere, Appendix A: Variations on Riemannian Flows, Pierre Molino, Riemannian Foliations, Springer, page 217 ↗,

     The object of this appendix is to give a summary of known results on 1-dimensional oriented Riemannian Foliations.

2. (music) Relating to the musical theories of German theorist Hugo Riemann, particularly his theory of harmony, which is characterised by a system of "harmonic dualism".
   • 1994, Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents, University of Chicago Press, page 7 ↗,

     And not only theory: most central European composers of this century were schooled in Riemannian doctrine of one type or another.

   • 2004, Jairo Moreno, Musical Representations, Subjects, and Objects: The Construction of Musical Thought in Zarlino, Descartes, Rameau, and Weber, Indiana University Press, page 1 ↗,

     Or take, for example, the rehabilitation by late-twentieth-century North American theorists of Riemannian Tonnetze as a means to navigate the voice-leading intricacies of much chromatic and post-chromatic music.

   • 2009, Marek Žabka, Generalized Tonnetz and Well-Formed GTS: A Scale Theory Inspired by the Neo-Riemannians, Elaine Chew, Adrian Childs, Ching-Hua Chuan (editors), Mathematics and Computation in Music, 2nd International Conference, Proceedings, Springer, page 286 ↗,

     The paper connects two notions originating from different branches of the recent mathematical music theory: the neo-Riemannian Tonnetz and the property of well-formedness from the theory of the generated scales.

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