Riemannian

English

Etymology

From Riemann +‎ -ian.

Adjective

Riemannian (not comparable)

(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”, in 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 ${\overline {g}}=dt^{2}+t^{2}g$ 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.
(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”, in 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.