Related papers: A note on Raki\'c duality principle for Osserman m…

We show that 4-dimensional Riemannian manifolds which satisfy the Raki\'c duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant), thus both conditions are equivalent.

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Raki\'c duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable…

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman…

Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.

In this note, we find a necessary condition on odd-dimensional Riemannian manifolds under which both of Sasakian structure and the generalised Ricci soliton equation are satisfied, and we give some examples.

We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the…

A condition of Osserman type, called $\phi$-null Osserman condition, is introduced and studied in the context of Lorentz globally framed $f$-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz…

Some little considerations concerning the application of the Theory of Dirichlet Forms to stocastic variational principle on riemannian manifolds are performed

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as another…

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded…

Let M be a Riemannian manifold and R its curvature tensor. For a unit vector X tangent to M at a point p, the Jacobi operator is defined by R_X = R(X, .) X$. The manifold M is called pointwise Osserman if, for every point p, the spectrum of…

In this paper, we study Jacobi operators associated to algebraic curvature maps (tensors) on lightlike submanifolds M. We investigate conditions for an induced Rie- mann curvature tensor to be an algebraic curvature tensor on M. We…

We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.

In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric…

In this note we formulate a condition for complete, connected and non-compact Riemannian manifolds which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.

In this work, we find an equation that relates the Ricci curvature of a riemannian manifold $M$ and the second fundamental forms of two orthogonal foliations of complementary dimensions, $\mathcal{F}$ and $\mathcal{F}^{\bot}$, defined on…

For a Riemannian manifold $M^n$ with the curvature tensor $R$, the Jacobi operator $R_X$ is defined by $R_XY = R(X,Y)X$. The manifold $M^n$ is called {\it pointwise Osserman} if, for every $p \in M^n$, the eigenvalues of the Jacobi operator…

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence…

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.