Local splitting of the tangent bundle with interesting properties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:24:31Z http://mathoverflow.net/feeds/question/93840 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93840/local-splitting-of-the-tangent-bundle-with-interesting-properties Local splitting of the tangent bundle with interesting properties Klaus Kröncke 2012-04-12T08:34:10Z 2012-04-15T11:57:24Z <p>Let $(M,g)$ be a Riemannian manifold and let $U\subset M$ be an open subset. Suppose that the tangent bundle over $U$ splits into two orthogonal distributions $TU=\mathcal{E}\oplus \mathcal{F}$.</p> <p>Is it possible that the two $C^{\infty}(U)$-bilinear maps </p> <p>\begin{align*}I:\mathcal{E}\times\mathcal{E}&amp;\to \mathcal{F}\ &amp;(X,Y)\mapsto pr_{\mathcal{F}}(\nabla_X Y) \end{align*}</p> <p>and</p> <p>\begin{align*}I:\mathcal{F}\times\mathcal{F}&amp;\to \mathcal{E}\ &amp;(X,Y)\mapsto pr_{\mathcal{E}}(\nabla_X Y) \end{align*}</p> <p>are both antisymmetric in $X$ and $Y$ without vanishing?</p> <p>If $\mathcal{E}$ and $\mathcal{F}$ both were integrable, both maps would be symmetric. So is this in some sense the most non-integrable way, distributions can be?</p> http://mathoverflow.net/questions/93840/local-splitting-of-the-tangent-bundle-with-interesting-properties/93890#93890 Answer by Robert Bryant for Local splitting of the tangent bundle with interesting properties Robert Bryant 2012-04-12T18:05:37Z 2012-04-15T11:57:24Z <p>Yes, this can happen. A little experimentation with the structure equations shows that there is a $3$-parameter family of homogeneous examples in dimension $4$: Let $c_1,\ldots,c_4$ be nonzero constants satisfying $c_1c_2=c_3c_4$, and consider the simply-connected $4$-dimensional Lie group $G$ that has a basis of left-invariant $1$-forms $\omega_1,\ldots,\omega_4$ that satisfy the structure equations \begin{aligned} d\omega_1 &amp;= 2c_1\ \omega_2\wedge\omega_3 + 2c_3\ \omega_4\wedge\omega_3\ ,\\ d\omega_2 &amp;= 2c_1\ \omega_3\wedge\omega_1 \ ,\\ d\omega_3 &amp;= 2c_2\ \omega_4\wedge\omega_1 + 2c_4\ \omega_2\wedge\omega_1\ ,\\ d\omega_4 &amp;= 2c_2\ \omega_1\wedge\omega_3 \ . \end{aligned}</p> <p>Now endow $G$ with the Riemannian metric $g$ for which the $\omega_i$ define an orthonormal coframing, let $e_1,\ldots,e_4$ be the dual ($g$-orthonormal) vector fields, and let $\mathcal{E}$ be the $2$-plane field spanned by $e_1$ and $e_2$ while $\mathcal{F}$ is the $2$-plane field defined by $e_3$ and $e_4$. </p> <p>One easily checks that this is an example of the desired type: If $\nabla$ is the Levi-Civita connection of this metric, then $$\nabla_{e_1}e_1\equiv0\ ,\quad\nabla_{e_1}e_2\equiv c_4e_3, \quad \nabla_{e_2}e_1\equiv-c_4e_3\ ,\quad\nabla_{e_2}e_2\equiv 0 \mod \mathcal{E}$$ and $$\nabla_{e_3}e_3\equiv0\ ,\quad\nabla_{e_3}e_4\equiv c_3e_1, \quad \nabla_{e_4}e_3\equiv-c_3e_1\ ,\quad\nabla_{e_4}e_4\equiv 0 \mod \mathcal{F},$$ as desired.</p> <p>There are non-homgeneous examples in dimension $4$ as well. A little more work with the structure equations shows that there exists a $4$-parameter family of examples of cohomogeneity $2$. (I don't know how many of these are complete.) If there is interest, I can give the structure equations of these examples as well.</p>