Curvature tensors of Hermitian elliptic spaces. (Russian) Zbl 0708.53028

Rozenfel’d, B. A.; Burtseva, T. A.; Dushina, N. V.; Kostrikina, L. P.; Yukhtina, T. I.

Tr. Geom. Semin. 20, 85-101 (1990).

As basis the authors take the formula obtained by P. A. Shirokov for the curvature tensor of symmetric Riemannian space \(V_{2n}\), which is isometric to the complex Hermitian elliptic space \[ {\mathbb{C}}\bar S_ n:\;R_{ijk\ell}=\frac{1}{r^ 2}(a_{ik}a_{j\ell}- a_{i\ell}a_{jk}+b_{ik}b_{j\ell}- b_{i\ell}b_{jk}+2b_{ij}b_{k\ell}), \] where \(a_{ij}\) is the metric tensor, and the antisymmetric tensor \(b_{ij}\) satisfies the condition \(b_{ij}b_{k\ell}a^{j\ell}=a_{ik}\). They obtain this formula using the general theory of symmetric spaces and by applying the apparatus of almost complex manifolds. Similar considerations are made for the space \({\mathbb{C}}_ q\bar S_ n\) over the algebra of double numbers, the quaternion and antiquaternion spaces \('{\mathbb{H}}\) \(\bar S_ n\) and \({\mathbb{H}}\bar S_ n\), for the octave and antioctave planes.

Reviewer: A.Shirokov

MSC:

53B35	Local differential geometry of Hermitian and Kählerian structures
53C35	Differential geometry of symmetric spaces

Keywords:

curvature tensor; Hermitian elliptic space; symmetric spaces