In a complete Riemannian manifold (M,g) if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of (M,g). In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds (M,g) admitting a real valued function u such that the hessian of u has atmost two eigenvalues −u and − 2 , under some mild hypothesis on (M,g)...

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. establish compactness properties of the set normalized quasi-convex functions and show local global comparison principles for twisted operators. then use Perron method to solve whose RHS involves an arbitrary probability measure, generalizing works Cheng-Yau, Delanoe, Caffarelli-Viaclovsky Hultgren-Onnheim. The intrinsi...

In this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of solitons that are realized as products, which know base spaces these products Ricci-Hessian type manifolds. We study latter class manifolds most appropriate setting to our results.

A self-similar Hessian (special Kähler) manifold is a $$(M,\nabla ,g)$$ endowed with an affine (holomorphic) homothetic vector field $$\xi $$ . Consider action of group G on ,g,\xi )$$ by isometries preserving such that acts the level set $$\{g(\xi ,\xi )=1\}$$ simply transitively. Then, we construct homogeneous conformally Kähler (hyper structure TM $$(T^*M)$$

Ambrose, Palais and Singer [6] introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath [23] extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Fréchet mani-folds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and disse...

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant curvature was obtained by B.-Y. Chen et al. Other particular cases inequalities a setting were given different authors. The objective the present article is to establish general curvature.

The geometry of Hessian manifolds is a fruitful branch physics, statistics, Kaehlerian and affine differential geometry. study inequalities for statistical submanifolds in constant curvature was truly initiated 2018 by Mihai, A. I. who dealt with Chen-Ricci Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. Ozel C. came the Casorati inequality same ambient space using algebraic technique. A...

Semi-supervised regression based on the graph Laplacian suffers from the fact that the solution is biased towards a constant and the lack of extrapolating power. Based on these observations, we propose to use the second-order Hessian energy for semi-supervised regression which overcomes both these problems. If the data lies on or close to a low-dimensional submanifold in feature space, the Hess...

We derive an a priori real Hessian estimate for solutions of large family geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent lower bound the right-hand side function. This improves estimates Székelyhidi [57] and additionally applies to with degenerate side. As application, we establish optimal C1,1 regularity envelopes (?,m)-subharmonic functions ...

In Proposition 4.1 a characterization is given of Hessian Rieman-nian structures in terms of a natural connection in the general linear group GL(n; R) + , which is viewed as a principal SO(n)-bundle over the space of positive deenite symmetric n n-matrices. For n = 2, Proposition 5.3 contains an interpretation of the curvature of a Hessian Riemannian structure at a given point, in terms of an u...