Here s is the scalar curvature, D2s the Hessian of s, ∆s = trD2s the Laplacian, and ◦ R the action of the curvature tensor R on symmetric bilinear forms, c.f. [B, Ch.4H] for further details. The equation (0.3) is just the trace of (0.2). It is obvious from the trace equation (0.3) that there are no non-flat R2 critical metrics, i.e. solutions of (0.2)-(0.3), on compact manifolds N ; this follow...

Abstract The approximation of probability measures on compact metric spaces and in particular Riemannian manifolds by atomic or empirical ones is a classical task complexity theory with wide range applications. Instead point we are concerned the supported Lipschitz curves. Special attention paid to push-forward Lebesgue unit interval such Using discrepancy as distance between measures, prove op...

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits metric be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, naturally appears as singular model in information geometry and related fields. A is locally accompanied with possibly multi-valued potential its dual, whose...

Abstract For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to function $E=\tfrac 12F^2$ . Under additional assumption that F is invariant with respect standard action $SO(k)\times SO(n-k)$ prove conjecture Laugwitz stated in 1965. Furthermore, describe all between such metrics, and Landsberg Unicorn C...

We interpret the setting for a Radon transform as a submanifold of the space of generalized functions, and compute its extrinsic curvature: it is the Hessian composed with the Radon transform. 1. The general setting. Let M and Σ be smooth finite dimensional manifolds. Let m = dim(M). A linear mapping R : C∞ c (M)→ C∞(Σ) is called a (generalized) Radon transform if it is given in the following w...

The problem of computing the best rank-(p, q, r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analy...

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère...

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of R steps of replica symmetry breaking. For the Parisi limit R → ∞ (continuum replica symmetry breaking) which is relevant for the manifold dimension D < 2, they are shown to be non negative. Resumé Les valeurs propres de la Hessienne, associée avec une varieté aleatoire, sont construites dans ...

In classical Morse theory the number and type (index) of critical points of a smooth function on a manifold are related to topological invariants of that manifold through the Morse inequalities. There the index of a critical point is the number of negative eigenvalues that the Hessian matrix has on that tangent plane. Here deenitions of \critical point" and \index" are given that are suitable f...