We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast freedom in choosing this submanifold. This freedom...

Let M̄ be a complex space form of constant holomorphic sectional curvature c and let M be an n-dimensional CR-submanifold of (n− 1) CR-dimension in M̄. ThenM has an almost contact metric structure (F,U ,u,g) (see Section 2) induced from the canonical complex structure of M̄. Hence on an n-dimensional CR-submanifold of (n− 1) CRdimension, we can consider two structures, namely, almost contact struc...

We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast freedom in choosing this submanifold. Th...

Local SU(3)-structures on an oriented submanifold of Spin(7)-manifold are determined and their types are characterized in terms of the shape operator and the type of Spin(7)-structure. An application to Bryant [5] and Calabi [10] examples is given. It is shown that the product of a Cayley plane and a minimal surface lying in a four-dimensional orthogonal Cayley plane with the induced complex st...

By a knot we mean a pair (S, M~) with M~ a smooth closed oriented submanifold of S (m^3). If such a knot is given and i:S-> gm+z j s t j ^ standard embedding, then one can isotope / in an essentially unique way (Lemma 1 below) to an embedding j :s -+S whose intersection with iS is M c S transversally. The «-fold cyclic branched cover of (S, iS) branched along (jS, M~) exists uniquely and is a m...

We use differential geometry techniques to estimate the class probability P (y = l|x) for learning both binary and multiclass plug-in classifiers. We propose a geometric regularization technique to find the optimal submanifold corresponding to the estimator of P (y = l|x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces fast osci...