Dirac operators are important geometric operators on a manifold. The Dirac operator DA on the four dimensional Euclidean space M = R is the order one differential operator whose square DA ◦ DA is the Euclidean Laplacian − ∑4 i=1 ∂ψ ∂xi . However, this is not possible unless we allow coefficients for this linear operator to be matrix-valued. Let M = R be the four dimensional Euclidean space with...

A Bertrand curve in the 4-dimensional Euclidean space is a whose first normal line same as of another curve. On other hand, Mannheim second or third In this paper, we investigate not only and curves regular curves, but also singular curves. As consider framed We give conditions for to become It well-known that do exist under condition space. However, even

The generating function of correlators of dual operators on the boundary of (A)dS4 space corresponding to the conformally coupled φ 4-model is obtained up to first order in the coupling constant by using the conformal map between massless scalar fields in (Euclidean) Minkowski space and conformally coupled scalars on (Euclidean anti) de Sitter space. Some exact classical solutions of the nonlin...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is (A) = 1 2 t(M) where M are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible fiel...

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is (A µ) αβ = 1 2 t µνλ (M νλ) αβ where M νλ are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of...

We generalize previous works on the Dirac eigenvalues as dynamical variables of the Euclidean gravity and N = 1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their v...