# 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.

#### Derived terms

* neo-Riemannian
* pseudo-Riemannian
* Riemannian geometry
* Riemannian manifold

#### Translations
### Noun

Riemannian (plural Riemannians)
* (mathematics) One who uses or supports the work of German mathematician Bernhard Riemann.

#### Translations
### See also

* Riemann
* Riemann integral

### Further reading

* (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