For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t \log p_t(x, y) \to −d^2(x, y)$ for $t \to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion $p_t(x, y) \sim t^{−5/4} \exp(−d^2(x, y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form $G_2$ of the exceptional Lie group, and is equal to $G_2$ for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split $G_2$. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in $G_2$ is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0 \gt \sigma_n K_{\rm max}$, where $\sigma_n \in (\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if $M^n(n \ge 3)$ is a compact manifold whose $(n − 2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n−2)}_{\rm min} \gt \tau_n(n −2)R_0$, where $\tau_n \in(\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.