A de Rham decomposition type theorem for contact sub-Riemannian manifolds Article Swipe

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that ( M , H , g ) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H . Suppose that there exists a point $$q\in M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> such that the holonomy group $$\Psi (q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> acts reducibly on H ( q ) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊕</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> into $$\Psi (q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊕</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊕</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> of H into sub-distributions $$H_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> . Unlike the Riemannian case, the distributions $$H_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are not integrable, however they induce integrable distributions $$\Delta _i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> on $$M/\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:math> , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊕</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊕</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> , and the latter decomposition of $$T(U/\xi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>U</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> induces the decomposition of $$U/\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:math> into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.