Submanifolds of constant positive curvature I

Topological obstructions to fatness

“…The set RE.ˇ/ D fy 2V 0 W rankˇy rankˇz; 8z 2V 0 g is clearly open and dense in V 0 . The following result is essentially contained in Moore [14]. Theorem 1 is now a consequence of the above Claim applied to the linear subspace V n H 2 .B; R/ spanned by the Chern classes of the bundle, ie by the pull back of H 2 .B T n / under the classifying map.…”

Euler characters and submanifolds of constant positive curvature

“…On the other hand, via Morse theory, it was possible to show that the only n-dimensional positively curved space form which immerses isometrically into E [(3/2)n] or conformally into E [(3/2)n−1] , where [x] denotes the largest integer ≤ x, is the standard n-sphere with constant curvature metric. (See Proposition 4, page 455 in [11] and the Corollary, page 90 in [10]. )…”

“…On the other hand, there is a rich theory of locally defined n-dimensional submanifolds of constant positive curvature in E 2n−1 . Indeed, it was shown in [11] that there are two local types of such submanifolds, those consisting of "nonumbilic points" and those consisting of "weak umbilics". Moreover, it can be shown that the ones containing nonumbilic points depend upon n(n − 1) functions of a single variable, in the sense of Cartan-Kähler theory.…”

“…The case where N = 2n − 1 is analyzed in [11]. In this case, we say that p is a weak umbilic if the second fundamental form at p satisfies (1) for some choice of e n+1 (p) ∈ N p M ; otherwise p is called a nonumbilic.…”

A counterexample to a conjecture on flat bilinear forms

“…We provide a counterexample to a conjecture on the dimension of the nullity of a flat symmetric bilinear form.The algebraic theory of flat bilinear forms was developed by J. D. Moore after the seminal work of E. Cartan on exterior orthogonal quadratic forms as a tool to treat the "rigidity problem" for submanifolds; see [5] and the references therein. An R-bilinear form β : V n ×V n → W p,q into a vector space endowed with an indefinite inner product of type (p, q) is said to be flat if…”