Riemannian

From Riemann +‎ -ian.

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, ${\displaystyle {\hat {A}}(M)}$ is represented by the closed differential form

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

     where ${\displaystyle R}$ is the Riemannian curvature of the metric ${\displaystyle g}$, regarded as an ${\displaystyle \textstyle \operatorname {End} (TM)}$-valued two form, and ${\displaystyle {\sqrt {\operatorname {det} }}}$ 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 ${\displaystyle {\overline {g}}=dt^{2}+t^{2}g}$ together with the symplectic form ${\displaystyle \omega }$ into a Kähler manifold. Then ${\displaystyle {\overline {g}}}$ and ${\displaystyle \omega }$ define a complex structure ${\displaystyle {\overline {\Phi }}}$. Alternatively, one could ask for a Riemannian metric ${\displaystyle g}$ on ${\displaystyle M}$ which would define a Kähler metric ${\displaystyle h}$ on ${\displaystyle {\mathcal {Z}}}$ 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”, 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.

• neo-Riemannian
• pseudo-Riemannian
• Riemannian geometry
• Riemannian manifold

Riemannian (plural Riemannians)

1. (mathematics) One who uses or supports the work of German mathematician Bernhard Riemann.

• Riemann
• Riemann integral

• (mathematics):
  • Bernhard Riemann on Wikipedia.Wikipedia
  • Riemannian geometry on Wikipedia.Wikipedia
  • Riemannian manifold on Wikipedia.Wikipedia
• (music):
  • Riemannian theory on Wikipedia.Wikipedia
  • Neo-Riemannian theory on Wikipedia.Wikipedia
  • Tonnetz on Wikipedia.Wikipedia