A metric deformation and the first eigenvalue of Laplacian on 1-forms

Abstract

Abstract. We search for a higher-dimensional analogue of Calabi's example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on S", n > 5, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on 1-forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an (Sn , some metric) to an (Sn~l x S1, some metric).

Geometrical interpretation of electromagnetism in a 5-dimensional manifold

“…Finally, THK thanks for anonymous acquaintances who gave encouragements. While a metric deformation is considered in mathematical literature [14,15], we introduce a formalism which can treat the change of manifold in a more direct and intuitive way. Before discussing our formalism, we give a brief remark about the metric deformation in each reference.…”

“…In [14], the considered deformation is only about the metric, not about the manifold itself, that the metric of a flat 2D manifold undergoes a deformation by a coordinate transformation on the 2D manifold. In [15], the deformation of a manifold itself is treated, but the considered deformation is to cut off a part of an original manifold and then replace it with some other same dimensional manifold possessing a different metric. These two are different from our formalism since the one used in this paper is a physical operation on a manifold itself, and it works as a continuous change of the manifold without cutting it off.…”

“…It is well known that equation (15) (C6) There are three second derivatives of x with respect to τ . We apply equation (15) From the metric and the inverse metric of the deformed 5D manifold given at section IV, the Christoffel symbols of the deformed 5D manifold can be expressed in terms of the Christoffel symbols of the deformed space-time and electromagnetic 4-potential.…”

“…It is well known that equation (15) requires (C.6) There are three second derivatives of x with respect to τ . We apply equation (15) (C.8) However, the left-hand-side is just equal to f (τ ) due to equation (C.4).…”

“…It is well known that equation (15) requires (C.6) There are three second derivatives of x with respect to τ . We apply equation (15) (C.8) However, the left-hand-side is just equal to f (τ ) due to equation (C.4). Therefore, (C.9) The second and the last term can be combined to give a term with the Christoffel symbols of the first kind: (C.10) Until here, the details of gαβ are never used.…”