Narrow Results

We consider a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with $(f,g,u,v,\lambda )$-structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with $(f,g,u,v,\lambda )$-structure under the condition when f is parallel.

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.

In a tangent bundle endowed with g-natural metric G we investigate submanifolds defined by a vector field given on a base manifold. We give a sufficient condition for a vector field on M to define totally geodesic submanifold in (TM, G). The parallel vector field is discussed in more detail.

This paper deals with the study of invariant submanifolds of generalized Sasakian-space-forms with respect to Levi-Civita connection as well as semi-symmetric metric connection. We provide an example of such submanifolds and obtain many new results including the necessary and sufficient conditions under which the submanifolds are totally geodesic. The Ricci solitons of such submanifolds are also studied.

In the present paper, we study the invariant submanifolds of $f$-Kenmotsu manifolds. Firstly, we show that any invariant submanifold of $f$-Kenmotsu manifold is again $f$-Kenmotsu manifold and minimal. Then, we give some characterizations of totally geodesic submanifolds of $f$-Kenmotsu manifolds. We study 3-dimensional submanifold and prove that a 3-dimensional submanifold of $f$-Kenmotsu manifold is totally geodesic if and only if it is invariant. Also, $\eta$-Ricci soliton is considered on invariant submanifold of $f$-Kenmotsu manifolds. Lastly, the non-trivial examples are constructed to verify some of our obtained results.

We show that for a surface $\mathrm{\Sigma }$, the subgraph of the pants graph determined by fixing a collection of curves that cut $\mathrm{\Sigma }$ into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made in [2] and has the implication that an embedded product of Farey graphs in any pants graph is totally geodesic. In addition, we show that a pants graph contains a convex $n$-flat if and only if it contains an $n$-quasi-flat.

The object of this paper is to study anti-invariant submanifolds of trans-Sasakian manifolds. We characterize such submanifolds on the basis of parallelism, semi-parallelism and pseudo parallelism of the second fundamental form of the submanifolds. We also characterize totally umbilical anti-invariant submanifolds of trans-Sasakian manifolds. Existence of Legendre curves on such submanifolds has been analyzed. Whether an anti-invariant submanifold of a trans-Sasakian manifold inherits local symmetry from ambient space is investigated here. Nature of Ricci soliton on anti-invariant submanifolds of trans-Sasakian manifolds has been characterized.