Compact Orientable Riemannian Manifold Research Articles

The existence of embedded G-invariant minimal hypersurface

For a compact connected Lie group G acting as isometries of cohomogeneity not equal to 0 or 2 on a compact orientable Riemannian manifold \(M^{n+1},\) we prove the existence of a nontrivial embedded G-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most \(n-7.\)

Estimates for the first eigenvalue of the drifting Laplacian on embedded hypersurfaces

For an $(n-1)$-dimensional compact orientable smooth metric measure space $\big(M,g,e^{-f}dv_{g}\big)$ embedded in an $n$-dimensional compact orientable Riemannian manifold $N$, we successfully give a lower bound for the first nonzero eigenvalue of the drifting Laplacian on $M$, provided the Ricci curvature of $N$ is bounded from below by a positive constant and the weighted function $f$ on $M$ satisfies two constraints.

Some results on harmonic and bi-harmonic maps

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications

It is known from [K. Yano and M. Kon, Structures on Manifolds (World Scientific, 1984)] that the integration of the Laplacian of a smooth function defined on a compact orientable Riemannian manifold without boundary vanishes with respect to the volume element. In this paper, we find out the some potential applications of this notion, and study the concept of warped product pointwise semi-slant submanifolds in cosymplectic manifolds as a generalization of contact CR-warped product submanifolds. Then, we prove the existence of warped product pointwise semi-slant submanifolds by their characterizations, and give an example supporting to this idea. Further, we obtain an interesting inequality in terms of the second fundamental form and the scalar curvature using Gauss equation and then, derive some applications of it with considering the equality case. We provide many trivial results for the warped product pointwise semi-slant submanifolds in cosymplectic space forms in various mathematical and physical terms such as Hessian, Hamiltonian and kinetic energy, and generalize the triviality results for contact CR-warped products as well.

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let \({\mathcal{D}}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of Lχ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h−p of the number field \({\mathbb Q(\sqrt {-p})}\) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h−p.

Rotation Numbers and Isometries

Using the notion of rotation set for homeomorphisms of compact manifolds, we define the rotation homomorphism of a connected compact orientable Riemannian manifold and apply it to prove that the dimension of the isometry group of a connected compact orientable Riemannian 3-manifold without conjugate points is not greater than its first Betti number. In higher dimensions the same is true under the additional assumption that the fundamental cohomology class of the manifold is a cup product of integral 1-dimensional classes.

Bochner's technique in the theory of Riemannian almost product structures

Let (M, g) be a compact orientable Riemannian manifold with positive definite metric g and globally defined Riemannian almost product structure (P, g), where P is a tensor field of type (i.i) such that p2 = id and g(P, P) = g. With the Riemannian almost product structure (P, g) there is associated a pair of orthogonal complementary distributions &H and.&V i corresponding to the eigenvalues of P equal, respectively, to -i and I. We denote by Pj, Rijke, and gke the components of the field P, the curvature tensor, and metric g with respect to a local coordinate system x ~, x 2, .... x n. In the expression for the scalar curvature R = gikgjeRijke we replace the contravariant components of the metric tensor g by the components pij = gjep~ of the structure tensor. We call the quantity H = pikpjeRijke obtained the scalar curvature of the Riemannian almost product structure. We call R (M) = ~ Rdv and N (7~f) = IM Ndv the global scalar curvatures of the Riemannian manifold (M~ g) and the Riemannian almost product structure (P~ g). A Riemannian almost product structure (P, g) is said to be integrable if the distributions A H and A V are integrable and thus there is given on (M, g) a pair of orthogonal complementary foliations. A foliation is said to be minimal (completely umbilical or completely geodesic) if each of its leaves is a minimal (respectively, completely umbilical or completely geodesic) submanifold of (M, g). One can prove the following theorem. THEOREM i. Let (M, g) be a compact, orientable, Riemannian manifold with globally defined Riemannian almost product structure. If ~(M) < R(M), then the Riemannian almost product structure does not define two orthogonal complementary minimal foliations on (M, g). If H(M) = R(M) such foliations can only be completely geodesic. A distribution A is said to be completely geodesic [3] if for any geodesic 7 of the Riemannian manifold (M, g) tangent at one of its points x to the plane A x, 7 is an integral curve of the distribution A. In other terminology such a distribution is called flat [4] since it is characterized by the vanishing of its second fundamental form.

Euler-Poincaré characteristic and higher order sectional curvature. I

The following long-standing conjecture of H. Hopf is well known. Let M M be a compact orientable Riemannian manifold of even dimension n ⩾ 2 n \geqslant 2 . If M M has nonnegative sectional curvature, then the Euler-Poincaré characteristic χ ( M ) \chi (M) is nonnegative. If M M has nonpositive sectional curvature, then χ ( M ) \chi (M) is nonnegative or nonpositive according as n ≡ 0 n \equiv 0 or 2 mod 4 2\bmod 4 . This conjecture for n = 4 n = 4 was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general n n under an extra condition on higher order sectional curvature, which holds automatically for n = 4 n = 4 . Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur’s theorem about the constancy of sectional curvature on a Riemannian manifold is extended.

CRITICAL POINTS OF THE MULTIDIMENSIONAL DIRICHLET FUNCTIONAL

The question of the existence of local minima is considered for the Dirichlet (energy) functional on spaces of mappings of one Riemannian manifold into another. In particular, it is shown that if the identity mapping of a compact irreducible homogeneous space onto itself has positive index, then any nonconstant harmonic mapping of an arbitrary compact orientable Riemannian manifold into such a space also has positive index.Bibliography: 15 titles.

On the second cohomology group of a pinched Riemannian manifold

Let M be a compact orientable Riemannian manifold of dimension even. We assume that the manifold can carry a metric, which is positively k-pinked. In the present paper some properties of the second cohomology group of such a manifold are obtained, when the number k is greater than a number which depends on the dimension of the manifold. These properties have some applications on the topological product of some special manifolds.

Studies on Riemannian Homogeneous Spaces

The purpose of the present paper is to give the details of the results announced in the C. R. notes [7], [8] and [9].In I which corresponds to [7], we shall develop a method for the study of affine transformations of a Riemannian manifold and prove Lemma 5 which is fundamental for the results in I and II. This part of our work has been motivated by a result of K. Yano [12] stating that the largest connected group of affine transformations of a compact orientable Riemannian manifold consists of isometries. Our method will have further applications to some other problems concerning or involving the connected group of isometries of a Riemannian manifold (for example, [10]).

On Riemannian manifolds of four dimensions

Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretic reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n ( > 4 ) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind. I t is the purpose of this note to give a study of a compact orientable Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study [2] . 2 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differential invariants constructed from a Riemannian metric previously given on the manifold. These two topological invariants have a linear combination which is the Euler-Poincare characteristic.