On the immersion of an 𝑛-dimensional manifold in 𝑛+1-dimensional Euclidean space

Abstract

Consider the subset of n + 1 n + 1 -dimensional Euclidean space swept out by the tangent hyperplanes drawn through the points of an immersed compact closed connected n-dimensional smooth manifold. If this is not all of the Euclidean space, then the manifold is diffeomorphic to a sphere, the immersion is an embedding, the image of the immersion is the boundary of a unique open starshaped set, and the set of points not on any tangent hyperplane is the interior of the kernel of the open starshaped set. A converse statement also holds.