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

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