This article introduces the Clairaut conformal Riemannian map. This notion includes the previously studied notions of Clairaut conformal submersion, Clairaut Riemannian submersion, and the Clairaut Riemannian map as particular cases, and is well known in the classical theory of surfaces. Toward this, we find the necessary and sufficient condition for a conformal Riemannian map $\varphi : M \to N$ between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth $s = e^f$. We show that the fibers of $\varphi $ are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, $-\nabla f$. Using this, we obtain a local splitting of M as a warped product and a usual product, if the horizontal space is integrable (under some appropriate hypothesis). We also provide some examples of the Clairaut conformal Riemannian maps to confirm our main theorem. We observe that the Laplacian of the logarithmic girth, that is, of f, on the total manifold takes the special form. It reduces to the Laplacian on the horizontal distribution, and if it is nonnegative, the universal covering space of M becomes a product manifold, under some hypothesis on f. Analysis of the Laplacian of f also yields the splitting of the universal covering space of M as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of M when f is a distance function. We also find the relationships between the total manifold and the fibers being symmetrical and, in particular, having constant sectional curvature, and from there, we compare their universal covering spaces, if fibers are also complete, provided f is a distance function. We also find a condition on the curvature tensor of the fibers to be semi-symmetric, provided that the total manifold is semi-symmetric and f is a distance function. In turn, this gives the warped product of symmetric, semi-symmetric spaces into two symmetric, semi-symmetric subspaces (under some hypothesis). Also if the Hessian or the Laplacian of the Riemannian curvature tensor fields is zero, or has a harmonic curvature tensor, then the fibers of $\varphi $ also satisfy the same property, if f is also a distance function. By obtaining Bochner-type formulas for Clairaut conformal Riemannian maps, we establish the relations between the divergences of the Ricci curvature tensor on fibers and horizontal space and the corresponding scalar curvature. We also study the horizontal Killing vector field of constant length and show that they are parallel under appropriate hypotheses. This in turn gives the splitting of the total manifold, if it admits a horizontal parallel Killing vector field and if the horizontal space is integrable. Finally, assuming that $\nabla f$ is a nontrivial gradient Ricci soliton on M, we prove that any vertical vector field is incompressible and hence the volume form of the fiber is invariant under the flow of the vector field.

A new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semi-invariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

We show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

First, we classify a real hypersurface of a non-flat complex space form with (i) semi-parallel T(=£ξg), and (ii) recurrent T. Next, we characterise a real hypersurface admitting the generalised η-Ricci soliton in a non-flat complex space form.

Weitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so-called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non-symmetric holonomy groups. We explicitly construct a basis of the space of Weitzenböck formulas. This classification allows us to find customized Weitzenböck formulas for applications such as eigenvalue estimates or Betti number estimates.

Starting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of ℂPn(4) as discovered by Opreaffi We show that, for n≥4, an n-dimensional Lagrangian submanifold in ℂPn(4) for which equality is attained at all points is necessarily minimal.

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $\left( A \right)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m > 2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m > 2$, on three real parameters along with an arbitrary Kähler–Einstein metric $h$ in complex dimension $m - 1$. We provide an explicit description of all these local-isometry types, for any given $h$. This result is derived from a more general local classification theorem for metrics admitting functions that we call special Kähler–Ricci potentials.

Si $M$ est une variété de dimension $n$, compacte non simplement connexe, on caractérise les métriques riemanniennes sur $M$ dont la fonction croissance a exactement deux singularités.

We characterize four-dimensional generalized complex forms and construct an Einstein and weakly *-Einstein Hermitian manifold with pointwise constant holomorphic sectional curvature which is not globally constant.