Definition of "Riemannian"

Similarly, A ^ ( M ) {\displaystyle {\hat {A}}(M)} is represented by the closed differential form A ^ ( M ) = det ( R / 2 sinh ⁡ ( R / 2 ) ) {\displaystyle {\hat {A}}(M)={\sqrt {\operatorname {det} }}\left({\frac {R/2}{\sinh(R/2)}}\right)} / where R {\displaystyle R} is the Riemannian curvature of the metric g {\displaystyle g} , regarded as an End ⁡ ( T M ) {\displaystyle \textstyle \operatorname {End} (TM)} -valued two form, and det {\displaystyle {\sqrt {\operatorname {det} }}} is the Pfaffian, which is an invariant polynomial defined on the Lie algebra of skew symmetric matrices in even dimensions.

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

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 g ¯ = d t 2 + t 2 g {\displaystyle {\overline {g}}=dt^{2}+t^{2}g} together with the symplectic form ω {\displaystyle \omega } into a Kähler manifold. Then g ¯ {\displaystyle {\overline {g}}} and ω {\displaystyle \omega } define a complex structure Φ ¯ {\displaystyle {\overline {\Phi }}} . Alternatively, one could ask for a Riemannian metric g {\displaystyle g} on M {\displaystyle M} which would define a Kähler metric h {\displaystyle h} on Z {\displaystyle {\mathcal {Z}}} via a Riemannian submersion.

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

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

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

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

1994, Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents, University of Chicago Press, page 7

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.

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

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.

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