Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Index parity of closed geodesics and rigidity of Hopf fibrations

Abstract: The diameter rigidity theorem of Gromoll and Grove [1987] states that a Riemannian manifold with sectional curvature ≥ 1 and diameter ≥ π/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane Caℙ. The reason that they were only able to recognize the cohomology ring of Caℙ is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion σ:?m→B b with connected fibers that is defined on the Euclidean sphere ?m is metrically congruent to a Hopf fibration unless possibly (m,b)=(15,8). We will rule out the exceptional cases in both theorems. Our argument relies on a rather unusal application of Morse theory. For that purpose we give a general criterion which allows to decide whether the Morse index of a closed geodesic is even or odd.

Manifolds of Riemannian metrics with prescribed scalar curvature

Abstract: THEOREM 2. Assume J * V 0 . Writing UtQ=(je a o\0 )U&9 J(\ is the disjoint union of closed submanifolds. REMARK. If d i m M = 2 , e^J=^" 8 , and if d i m M = 3 , the hypothesis that 1F*J£0 can be dropped. The proof of Theorem 1 also allows us to conclude that a solution h of the linearized equations DR(g0) • h=0 is tangent to a curve of exact solutions of R(g)=p through a given solution g0, provided p is not a constant ^ 0 . In the terminology of [4] we say the equation R(g)=p is linearization-stable at g0. From Theorem 3 below the equation R(g)=0 is still linearization-stable about a solution g0 provided Ric(g0) is not identically zero. For the singular case p = 0 , Theorem 2 incorporates an isolation theorem inspired by the work of Brill and Deser [2], namely, that the flat metrics are isolated solutions of R(g)=0. As a corollary one has: If g(t) is a

Sharp estimates and the Dirac operator

Abstract: This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric $g$ on $S^n$ that strictly dominates the standard metric $g_0$ must have somewhere scalar curvature strictly less than that of $g_0$ . More generally, if $M$ is any compact spin manifold of dimension $n$ which admits a distance decreasing map $f:M \rightarrow S^n$ of non-zero degree, then either there is a point $x \in M$ with normalized scalar curvature $\tilde{\kappa}(x)< 1$ , or $M$ is isometric to $S^n$ . The distance decreasing hypothesis can be replaced by the weaker assumption $f$ is contracting on $2$ -forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by $k$ -forms with $k \geq 3$ .

Parametric representation of a surface pencil with a common line of curvature

Abstract: Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed.