In this paper we study harmonic maps relative to α-connections, and not always relative to Levi-Civita connections, on statistical manifolds. In particular, harmonic maps on α-conformally equivalent statistical manifolds are discussed, and conditions for harmonicity are given by parameters α and dimensions n. As the application we also describe harmonic maps between level surfaces of a Hessian ...

In this paper, we consider a class of Hessian type equations which include the $(n-1)$ Monge-Amp\`{e}re equation on Riemannian manifolds. The \emph{a priori} $C^2$ estimates and existence solutions are established.

In this paper, we derive an a priori second order estimate for solutions which are in Γk+1 cone to class of complex Hessian equations with both sides the equation depending on gradient compact Hermitian manifolds.

In this paper, we establish a priori estimates for solutions of general class fully non-linear equations on compact almost Hermitian manifolds. As an application, solve the complex Hessian equation and Monge–Ampère $$(n-1)$$ -plurisubharmonic functions in setting.

We derive second order estimates for \begin{document}$ \chi $\end{document}-plurisubharmonic solutions of complex Hessian equations with right hand side depending on the gradient compact Hermitian manifolds.

The Riemannian trust-region algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be cha...

We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps. We present and prove conditions on the manifold that are necessary for the s...