here, a finsler manifold $(m,f)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. certain subspaces of the tangent spaces of $m$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. it is shown that if the dimension of foliation is constant, then the distribution is involutive a...

Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...

We show that the number of fundamentally different m-colorings of a knot K depends only on the m-nullity of K, and develop a formula for the number of such colorings. We also determine the m-colorability and m-nullity of any (p, q, r) pretzel knot, and therefore determine the number of fundamentally different m-colorings for any (p, q, r)

Let G = (V, E) be a graph with V = {1, 2, . . . , n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A = [ai,j ] with ai,j = 0 if i = j and i and j are non-adjacent, ai,j = 0 if i = j and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-defi...

Let G = (V, E) be a graph with V = {1, 2, . . . , n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A = [ai,j ] with ai,j = 0 if i = j and i and j are non-adjacent, ai,j = 0 if i = j and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-defi...

We prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakianmanifold M of rank l is at least equal to 2l−1 provided thatM has an isolated closed characteristic. The result is then used to provide some examples ofK-contact manifolds which are not Sasakian. On a closed, 2n+ 1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension o...

The aim of this paper is to characterize 3-dimensional almost Kenmotsu manifolds with ξ belonging to the (k, μ)′-nullity distribution and h′ 6= 0 satisfying certain geometric conditions. Finally, we give an example to verify some results.