In scientific computing and machine learning applications, matrices more general multidimensional arrays (tensors) can often be approximated with the help of low-rank decompositions. Since tensors fixed rank form smooth Riemannian manifolds, one popular tools for finding approximations is to use optimization. Nevertheless, efficient implementation gradients Hessians, required in optimization al...

5 1. Prologue 5 2. INTRODUCTION 8 2.1. The general concerns 8 2.2. The geometry of finite dimensional Cartan-Lie groups and abstract Lie groups 9 2.3. The information geometry of complex systems 9 2.4. The gauge group 9 2.5. The overview of the main results: Solutions to some problems EX(S) 10 2.6. Geometry of Lie groups 13 2.7. A comment on Figues 17 3. BASIC NOTIONS. 18 3.1. Cartan-Lie groups...

My work focuses on the geometry and differential equations invariant under groups of affine and projective motions (in R and RP respectively). In particular, affine differential geometry, the study of properties of hypersurfaces in R which are invariant under affine volume-preserving motions, informs most of my work. Affine differential geometry is an old subfield of geometry, with Blaschke mak...

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...

We generalize the classical Bochner formula for heat flow on evolving manifolds $$(M,g_{t})_{t \in [0,T]}$$ to an infinite-dimensional martingales parabolic path space $$P{\mathcal {M}}$$ of space-time $${\mathcal {M}} = M \times [0,T]$$ . Our new and inequalities that follow from it are strong enough characterize solutions Ricci flow. Specifically, we obtain characterizations in terms space. a...

Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.

The object of this paper is to study $(epsilon)$-Lorentzian para-Sasakian manifolds. Some typical identities for the curvature tensor and the Ricci tensor of $(epsilon)$-Lorentzian para-Sasakian manifold are investigated. Further, we study globally $phi$-Ricci symmetric and weakly $phi$-Ricci symmetric $(epsilon)$-Lorentzian para-Sasakian manifolds and obtain interesting results.

commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.