### ರಿಮ್ಯಾನಿಯನ್

• Definition

of or relating to Riemann's non-Euclidean geometry

ರೀಮನ್‌ನ ಯೂಕ್ಲಿಡಿಯನ್ ಅಲ್ಲದ ರೇಖಾಗಣಿತದ ಅಥವಾ ಸಂಬಂಧಿಸಿದೆ

### ರಿಮ್ಯಾನಿಯನ್

• Definitions

1. Of or relating to the work, or theory developed from the work, of German mathematician Bernhard Riemann, especially to Riemannian manifolds and Riemannian geometry.

ಜರ್ಮನ್ ಗಣಿತಜ್ಞ ಬರ್ನ್‌ಹಾರ್ಡ್ ರೀಮನ್‌ರ ಕೆಲಸದಿಂದ ಅಭಿವೃದ್ಧಿಪಡಿಸಿದ ಕೃತಿ ಅಥವಾ ಸಿದ್ಧಾಂತಕ್ಕೆ ಸಂಬಂಧಿಸಿದೆ, ವಿಶೇಷವಾಗಿ ರೈಮನ್ನಿಯನ್ ಮ್ಯಾನಿಫೋಲ್ಡ್‌ಗಳು ಮತ್ತು ರೈಮನ್ನಿಯನ್ ರೇಖಾಗಣಿತಕ್ಕೆ

• Examples:
1. 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.

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

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

• 2. Relating to the musical theories of German theorist Hugo Riemann, particularly his theory of harmony, which is characterised by a system of "harmonic dualism".

ಜರ್ಮನ್ ಸಿದ್ಧಾಂತಿ ಹ್ಯೂಗೋ ರೀಮನ್ ಅವರ ಸಂಗೀತ ಸಿದ್ಧಾಂತಗಳಿಗೆ ಸಂಬಂಧಿಸಿದೆ, ವಿಶೇಷವಾಗಿ ಅವರ ಸಾಮರಸ್ಯದ ಸಿದ್ಧಾಂತ, ಇದು "ಹಾರ್ಮೋನಿಕ್ ದ್ವಂದ್ವವಾದ" ವ್ಯವಸ್ಥೆಯಿಂದ ನಿರೂಪಿಸಲ್ಪಟ್ಟಿದೆ.

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

2. 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.

3. 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.

• Synonyms

Riemannian geometry (ರೈಮನ್ನಿಯನ್ ರೇಖಾಗಣಿತ)

pseudo-Riemannian (ಹುಸಿ-ರೀಮ್ಯಾನಿಯನ್)

Riemannian manifold (ರಿಮ್ಯಾನಿಯನ್ ಮ್ಯಾನಿಫೋಲ್ಡ್)

neo-Riemannian (ನವ-ರೀಮ್ಯಾನಿಯನ್)