Skip to main content Accessibility help
×
Home
Hostname: page-component-5bf98f6d76-v92w2 Total loading time: 0.273 Render date: 2021-04-20T13:38:26.203Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Jacobi's elliptic functions and Lagrangian immersions

Published online by Cambridge University Press:  14 November 2011

Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, U.S.A. e-mail: bychen@math.msu.edu
Corresponding
E-mail address:

Abstract

First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below.

References

1Bowman, F.. Introduction to Elliptic Functions with Applications (New York: John Wiley, 1961).Google Scholar
2Byrd, P. F. and Friedman, M. D.. Handbook of Elliptic Integrals for Engineers and Scientists (Berlin: Springer, 1971).CrossRefGoogle Scholar
3Chen, B. Y.. Geometry of Submanifolds (New York: M. Dekker, 1973).Google Scholar
4Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, L.. Totally real submanifolds of ℂPn satisfying a basic equality. Arch. Math. (Basel) 63 (1994), 553–64.CrossRefGoogle Scholar
5Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, L.. An exotic totally real minimal immersion of S 3 in ℂP 3 and its characterisation. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 153–65.CrossRefGoogle Scholar
6Chen, B. Y., Ludden, G. D. and Montiel, S.. Real submanifolds of a Kaehler manifold. Algebra Groups Geom. 1(1984), 176212.Google Scholar
7Chen, B.-Y. and Oguie, K.. On totally real submanifolds. Trans. Amer. Math. Soc. 193 (1974), 257–66.CrossRefGoogle Scholar
8Dillen, F. and Nolker, S.. Semi-parallelity, multi-rotation surfaces and the helix-property. J. Reine Angew. Math. 435 (1993), 3363.Google Scholar
9Hiepko, S.. Eine innere Kennzeichung der verzerrten Produkte. Math. Ann. 241 (1979), 209–15.CrossRefGoogle Scholar
10Nölker, S.. Isometric immersions of warped products. Differential Geom. Appl. (to appear).Google Scholar
11O'Neill, B.. Semi-Riemannian Geometry with Applications to Relativity (New York: Academic Press, 1983).Google Scholar
12Reckziegel, H.. Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. Global Differential Geometry and Global Analysis (1984), Lecture Notes in Mathematics 1156, pp. 264–79 (Berlin: Springer, 1985).Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 32 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 20th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Jacobi's elliptic functions and Lagrangian immersions
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Jacobi's elliptic functions and Lagrangian immersions
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Jacobi's elliptic functions and Lagrangian immersions
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *