Invariant affine connections on odd-dimensional spheres

Abstract

A Riemann–Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann–Cartan space is introduced in a natural way. The paper is focused on the case of the odd-dimensional spheres $\mathbb S^{2n+1}$ endowed with their canonical Riemannian round metrics and viewed as homogeneous spaces of the special unitary groups. The classical Nomizu’s Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the spheres $\mathbb S^{2n+1}$ into homogeneous Riemann–Cartan spaces. The expressions of such connections as covariant derivatives are given by means of several invariant tensors: the ones of the usual Sasakian structure of the sphere; an invariant 3-differential form coming from a 3-Sasakian structure on $\mathbb S^7$; and the involved ones in the almost contact metric structure of $\mathbb S^5$ provided by its natural embedding into the nearly Kähler manifold $\mathbb S^6$. Furthermore, the invariant connections sharing geodesics with the Levi-Civita one have also been completely described. Finally, $\mathbb S^3$ and $\mathbb S^7$ are characterized as the unique odd-dimensional spheres which admit nontrivial invariant connections satisfying an Einstein-type equation.

Appendix

Here is a summary of the results of this work:

Invariant connections on odd-dimensional spheres
	Invariant	7	$\leftrightarrow \langle \{\alpha _{1},\, \alpha _\mathbf{{i}},\,\beta _{1},\,\beta _\mathbf{{i}},\,\gamma _{1},\, \gamma _\mathbf{{i}},\,\delta _1\}\rangle$
$\mathbb S^{2n+1}$	Metric	3	$\nabla ^{g}_{X}Y+s_1(\Phi (X,Y)\,\xi +\eta (Y)\psi (X)) + s_2(g(X,Y)\,\xi -\eta (Y)X)+s_3\eta (X)\psi (Y)$
	Skew-torsion	1	$\nabla ^{g}_{X}Y+s_1T^{c}(X,Y)$
	$\nabla$-Einstein	Point	$\nabla ^{g}_{X}Y$
	Invariant	9	$\leftrightarrow \langle \{\alpha _{1},\, \alpha _\mathbf{{i}},\,\beta _{1},\,\beta _\mathbf{{i}},\,\gamma _{1},\, \gamma _\mathbf{{i}},\,\varepsilon _{1},\,\varepsilon _\mathbf{{i}},\,\delta _1\}\rangle$
$\mathbb S^{7}$	Metric	5	$\nabla ^{g}_{X}Y+s_1\, (\Phi _1(X,Y)\,\xi _1+\eta _1(Y)\psi _1(X) ) +s_2\, \eta _1(X)\psi _1(Y)+ s_3\, \nabla ^{g}_{X}\psi _1 + s_4 \, \Theta (X,Y) +s_5 \, \tilde{\Theta }(X,Y)$
	Skew-torsion	3	$\nabla ^{g}_{X}Y+ s_1 T^{c}(X,Y) + s_4 \, \Theta (X,Y) + s_5 \, \tilde{\Theta }(X,Y)$
	$\nabla$-Einstein	Cone	$s_4^2+s_5^2=s_1^2$
	Invariant	13	$\leftrightarrow \langle \{\alpha _{1},\, \alpha _\mathbf{{i}},\,\beta _{1},\,\beta _\mathbf{{i}},\,\gamma _{1},\, \gamma _\mathbf{{i}},\,\hat{\alpha }_{1},\, \hat{\alpha }_\mathbf{{i}},\,\hat{\beta }_{1},\,\hat{\beta }_\mathbf{{i}},\,\hat{\gamma }_{1},\, \hat{\gamma }_\mathbf{{i}},\,\delta _1\}\rangle$
$\mathbb S^{5}$	Metric	7	$\nabla ^{g}_{X}Y+s_1(\Phi (X,Y)\,\xi +\eta (Y)\psi (X)) + s_2\eta (X)\psi (Y)+ s_3(\Phi (\widehat{\psi }(X),Y)\,\xi +\eta (Y)\psi (\widehat{\psi }(X))) +s_4\eta (X)\psi (\widehat{\psi }(Y)) +s_5\eta (X)\widehat{\psi }(Y)+ s_6(g(\widehat{\psi }(X),Y)\,\xi -\eta (Y)\widehat{\psi }(X))+s_7 \nabla ^{g}_{X}\psi$
	Skew-torsion	3	$\nabla ^{g}_{X}Y +s_1\big (\Phi (\widehat{\psi }(X),Y)\,\xi -\eta (X)\psi (\widehat{\psi }(Y))+\eta (Y)\psi (\widehat{\psi }(X))\big )+s_2 T^{c}(X,Y) +s_3\big ( g(\widehat{\psi }(X),Y)\,\xi +\eta (X)\widehat{\psi }(Y)-\eta (Y)\widehat{\psi }(X)\big )$
	$\nabla$-Einstein	Point	$\nabla ^{g}_{X}Y$
	Invariant	27	$\nabla ^{g}_{X}Y+\sum s_{ijk}E_i^{\flat }(X)E_j^{\flat }(Y)E_k$
$\mathbb S^{3}$	Metric	9	$\nabla ^{g}_{X}Y+\sum s_{ij}E_i^{\flat }(X)(E_{j}^{\flat }(Y)E_{j+1}-E_{j+1}^{\flat }(Y)E_{j} )$
	Skew-torsion	1	$\nabla ^{g}_{X}Y+s_1T^{c}(X,Y)$
	$\nabla$-Einstein	Line	$\nabla ^{g}_{X}Y+s_1 T^{c}(X,Y)$,

where $T^{c}$ denotes the torsion tensor of the characteristic connection of the Sasakian manifold given in Example 4.11(iii).