Twistor Theory for Riemannian Symmetric Spaces

Mathematical Nanotechnology: Quantum Field Intentionality

“…If we consider the algebra e , that is to say, find their local application in their algebra , then we can access to the quantum zone of the nullity where the temporal effects are annulled. Then in , [3]. Then supposing that the field X can control under finite actions like the described for , and under the established principle, we can execute an action on a microstructure always and when the sum of the actions of all the particles is major than their algebraic sum (to give an order to only one particle so that the others continue it).…”

“…Likewise if we choose a geodesic t  , of the field X, on which an action will be applied , measurable in the Lagrange ambience, we have that their execution is as given in (3). The action must be realized in form supported along the object to which is required apply the transformation due to X.…”

“…, is their action due to a field of particles X, and is a spilled intention defined by (3) showing the length and breadth of the space , such that to satisfy (5), for all the possible trajectories that fill , we have M …”

Higher order symmetric spaces and the roots of the identity in a Lie group

“…The structure and classification of k-symmetric manifolds have been extensively studied in [5] and it has been known for a long time (cf. [2]) that every 2-symmetric manifold of the form G/G τ can be one-to-one immersed into the Lie group G, by associating gτ (g −1 ) with the class of an element g ∈ G, a fact that can be generalized trivially to k-symmetric spaces. We will prove, more precisely, that this one-to-one immersion is always an embedding, and hence that the k-symmetric space G/G τ admits a model Orb that is a submanifold of G. This fact was established in [1], for the special case where G is compact, where the name "Cartan embedding" is used (in fact, Burstall uses the inverse τ (g)g −1 instead of gτ (g −1 ) but this makes no essential difference).…”

“…Every k-symmetric space is reductive in a canonical way and, as such, it has a canonical connection, and it is known [2] that, for k = 2, its embedding into the Lie group G, considered with its symmetric connection, is totally geodesic. This fact led us to compute, for general k, the Hessian of the inclusions of r k (G) and Orb into G.…”