Riemannian
Language: English
Part of speech: adj
Etymology: From Riemann + -ian.
Senses: examples: text: where R is the Riemannian curvature of the metric g, regarded as an operatorname End(TM)-valued two form, and √ is the Pfaffian, which is an invariant polynomial defined on the Lie algebra of skew symmetric matrices in even dimensions.
ref: 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, ̂A(M) is represented by the closed differential form ̂A(M)=√((R/2)/(sinh (R/2)))
text: 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 #92;overlineg#61;dt²#43;t²ᵍ together with the symplectic form #92;omega into a Kähler manifold. Then #92;overlineg and #92;omega define a complex structure #92;overline#92;Phi. Alternatively, one could ask for a Riemannian metric g on M which would define a Kähler metric h on #92;mathcalZ via a Riemannian submersion.
ref: 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:
text: 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.
topics: mathematics
sciences
glosses: Of or relating to the work, or theory developed from the work, of German mathematician Bernhard Riemann, especially to Riemannian manifolds and Riemannian geometry.
examples: text: And not only theory: most central European composers of this century were schooled in Riemannian doctrine of one type or another.
ref: 1994, Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents, University of Chicago Press, page 7:
text: 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.
ref: 2004, Jairo Moreno, Musical Representations, Subjects, and Objects: The Construction of Musical Thought in Zarlino, Descartes, Rameau, and Weber, Indiana University Press, page 1:
text: 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.
ref: 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:
topics: entertainment
lifestyle
music
glosses: Relating to the musical theories of German theorist Hugo Riemann, particularly his theory of harmony, which is characterised by a system of "harmonic dualism".
Derived terms: neo-Riemannian
pseudo-Riemannian
Riemannian geometry
Riemannian manifold