The complex Monge–Ampère (CMA) equation in a two-component form is treated as bi-Hamiltonian system. I present explicitly the first nonlocal symmetry flow each of two hierarchies this An invariant solution CMA with respect to these symmetries constructed, which, being noninvariant usual sense, does not undergo reduction number independent variables. also construct corresponding four-dimensional...

Finding diffeomorphism-invariant observables to characterize the properties of gravity and spacetime at Planck scale is essential for making progress in quantum gravity. The holonomy Wilson loop Levi-Civita connection are potentially interesting ingredients construction curvature observables. Motivated by recent developments nonperturbative gravity, we establish new relations three four dimensi...

Let ν be a unit normal vector field along M . Notice that ν : M −→ S satisfies that 〈ν(m),m〉 = 0. For any tangent vector v ∈ TmM , m ∈ M , the shape operator A is given by A(v) = −∇̄vν, where ∇̄ denotes the Levi Civita connection in S. For every m ∈ M , A(m) defines a linear symmetric transformation from TmM to TmM ; the eigenvalues of this transformation are known as the principal curvatures of ...

We prove the well-posedness of initial boundary value problem for Einstein equations with sole condition requirement that timelike is totally geodesic. This provides first result this specific geometric and setting which uniqueness in original sense Friedrich holds problem. Our proof relies on ADM system vacuum equations, formulated respect to a parallelly propagated orthonormal frame along geo...

We consider non-degenerate graph immersions into affine space An+1 whose cubic form is parallel with respect to the Levi-Civita connection of metric. There exists a correspondence between such and pairs (J,?), where J an n-dimensional real Jordan algebra ? trace on J. Every immersion can be extended complete symmetric covering maximal connected component zero in set quasi-regular elements It im...

If $\psi\colon M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $$ \mathcal{F}\_m(\psi) = \int\_M 1 + |\nabla^m \nu |^2 , d\mu, where $\nu$ local unit normal vector along $\psi$, $\nabla$ Levi-Civita connection of Riemannian manifold $(M,g)$, with $g$ pull-back metric induced by immersion and $\mu$ associated volume measure. We prove that if $m>\lflo...