English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannianmetric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ thecenter of $\mathfrak{g}$. By using an orthonormal basis adapted to thesplitting  $\mathfrak{g}=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])\oplusO^+\oplus (Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp)\oplus  O^-$, where $O^+$ (resp. $O^-$) is the orthogonal of$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]$ in$[\mathfrak{g},\mathfrak{g}]$ (resp. is the orthogonal of$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp$ in$[\mathfrak{g},\mathfrak{g}]^\perp$), we show that the signature of the Riccioperator of $(G,h)$ is determined by the dimensions of the vector spaces$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}],$$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp$ and the signature of asymmetric matrix of order$\dim[\mathfrak{g},\mathfrak{g}]-\dim(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])$.This permits to associate to $G$ a subset $\mathbf{Sign}(\mathfrak{g})$ of$\mathbf{N}^3$ depending only on the Lie algebra structure, easy to compute andsuch that, for any left invariant Riemannian metric on $G$, the signature ofits Ricci operator belongs to $\mathbf{Sign}(\mathfrak{g})$. We show also thatfor any nilpotent Lie group of dimension less or equal to 6,$\mathbf{Sign}(\mathfrak{g})$ is actually the set of signatures of the Riccioperators of all left invariant Riemannian metrics on $G$. We give also somegeneral results which support the conjecture that the last result is true inany dimension.

The signature of the Ricci curvature of left-invariant Riemannian metrics on nilpotent Lie groups