Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate...

On the predual of a von Neumann algebra, we define a dif-ferentiable manifold structure and affine connections by embeddings into non-commutative L p –spaces. Using the geometry of uniformly convex Ba-nach spaces and duality of the L p and L q spaces for 1/p + 1/q = 1, we show that we can introduce the α-divergence, for α ∈ (−1, 1), in a similar manner as Amari in the classical case. If restric...

In this note I speculate on connections between the noncommutative geometry approach to the standard model on one side, and the internal space coming from strings on the other. The standard model in noncommutative geometry is described via the spectral action. I argue that an internal noncommutative manifold compactified at the renormalization scale, could give rise to the almost commutative ge...

The loop homology of a closed orientable manifoldM of dimensionm is the commutative algebra H∗(LM) = H∗+m(LM) equipped with the Chas-Sullivan loop product. Here LM denotes the free loop on M . There is also a homomorphism of graded algebras I from H∗(LM) to the Pontryagin algebra H∗(ΩM). The primary purpose of this paper is to study the loop homology and the homomorphism I, when M is a fibered ...

In this paper I discuss connections between the noncommutative geometry approach to the standard model on one side, and the internal space coming from strings on the other. The standard model in noncommutative geometry is described via the spectral action. I argue that an internal noncommutative manifold compactified at the renormalization scale, could give rise to the almost commutative geomet...

Since Isaac Newton the understanding of physical world is more and complex. The Euclidean space three dimensions , independent time replaced in Enstein’s vision by Lorentzian space-time at first, then four manifold to unify matter. String theorists add make their theory consistent. Complex topological invariants which characterize different kind spaces are developed. Space discretized quantum s...

The present article serves the purpose of pursuing Geometrization of heat flow on volumetrically isothermal manifold by means of RF approach. In this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of Riemannian manifold and fundamental properties of thermodynamic systems. By making use of the notions of various curva...

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...

We present a candidate supergravity solution for a stacked configuration of stable non-BPS D-branes in Type II string theory compactified on T /Z2. This gives a supergravity description of nonabelian tachyon condensation on the brane worldvolume. BROWN-HET-1263 IC/2001/27

In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...