On Morse theory for manifolds with cross products

We consider the finite-dimensional Morse theory for closed Riemannian manifolds equipped with the vector cross product on the tangent bundle. These are, for example, $G_2$-manifolds. Under some conditions toric actions generate the Morse–Bott function, whose gradient trajectories are explicit. This allows us to construct the Morse–Bott complex and calculate the real cohomology ring of the manifold.