On Frenet frames of complex submanifolds in complex projective spaces
type: Item
instance of: scholarly article
title: On Frenet frames of complex submanifolds in complex projective spaces
zbMATH Open document ID: 0582.32036
DOI: 10.1215/S0012-7094-84-05109-3
author: Hsin-Sheng Tai
published in: Duke Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The main results of this paper concern the curvature of a certain osculating bundle of complex projective manifolds. The author obtains a formula which generalizes a formula by Weyl for holomorphic curves - a so-called unintegrated Plücker formula. As a corollary he obtains the following characterization: The Veronese surfaces is the only surface in \({\mathbb{P}}^ 5\) whose second order osculating spaces all have the maximal dimension 5. This result has been generalized to a characterization of n-fold Veronese embeddings \({\mathbb{P}}^ r\to {\mathbb{P}}^ N,\) \(N=\left( \begin{matrix} n+r\\ r\end{matrix} \right)-1,\) by the author, \textit{W. Fulton}, \textit{S. Kleiman}, and the reviewer in Bull. Soc. Math. Fr. 113, 205-210 (1985).
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 51M35
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 53A17
zbMATH DE Number: 3931406
zbMATH Keywords: Frenet frame
zbMATH Keywords: Kähler form
zbMATH Keywords: Chern form
zbMATH Keywords: osculating metric
zbMATH Keywords: Grassmann
zbMATH Keywords: polar class
zbMATH Keywords: complex submanifolds in complex projective spaces
zbMATH Keywords: curvature
zbMATH Keywords: unintegrated Plücker formula
zbMATH Keywords: Veronese surfaces
zbMATH Keywords: osculating spaces
zbMATH Keywords: Veronese embeddings
reviewed by: Ragni Piene
MaRDI profile type: Publication
cites work: Characteristic classes of Hermitian manifolds
cites work: Q5568505
cites work: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry
cites work: Q4195061
cites work: Polar classes of singular varieties
cites work: Differential geometry of higher order
cites work: Q5801760
cites work: The Arithmetical Invariants of Algebraic Loci
cites work: Meromorphic Functions and Analytic Curves. (AM-12)
full work available at URL: s0012-7094-84-05109-3
OpenAlex ID: W1495614079
description: scientific article; zbMATH DE number 3931406