Download Coulomb Frames in the Normal Bundle of Surfaces in Euclidean by Steffen Fröhlich PDF

By Steffen Fröhlich

This e-book is meant for complicated scholars and younger researchers attracted to the research of partial differential equations and differential geometry. It discusses trouble-free thoughts of floor geometry in higher-dimensional Euclidean areas, specifically the differential equations of Gauss-Weingarten including quite a few integrability stipulations and corresponding floor curvatures. It contains a bankruptcy on curvature estimates for such surfaces, and, utilizing effects from power conception and harmonic research, it addresses geometric and analytic how you can identify the lifestyles and regularity of Coulomb frames of their common bundles, which come up as severe issues for a sensible of overall torsion.

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Additional info for Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces: Topics from Differential Geometry and Geometric Analysis of Surfaces

Example text

Let a minimal surface X W B ! X0 / 12 Proof. p; '/ with geodesic polar coordinates . ; '/ 2 Œ0; r Œ0; 2 : With the area element P . t. this coordinate system takes the form dsP2 D d 2 C P . ; '/ d'; with smooth P . ; '/ > 0 for all . 0; r lim P . 0C Œ0; 2 / satisfying @ p P . g. to Blaschke and Leichtweiß [12]. In particular, with the geodesic curvature Äg of the surface, the integral formula of Gauß–Bonnet gives Zr Z Z2 p p Äg . ; '/ P . ; '/ d' C K. ; '/ P . ; '/ d d' D 2 : 0 For curves with 0 0 D const it holds p @ p Äg .

R4 is parallel in the normal bundle then it has constant length. If additionally H 6D 0; then it holds S Á 0 for the scalar curvature of the normal bundle. Proof. The identities @? ui H D n X D1 H ;ui N C n X #D1 ? @v T1;1 D H2 S W 2 H2;v T1;1 2 H2 @v T1;1 2 H2 @v T1;1 2 @u T1;2 / and analogously 0 D H1 S W: Therefore, either X is a minimal immersion with H Á 0; or if not then it is a surface with mean curvature vector of constant length greater than zero and with flat normal bundle. The statement is proved.

T #;2 T#;1 ! L j;kD1 for ; ! ;k2 /g jk 20 1 Surface Geometry Both sides of these identities clearly vanish identically if n D 1: We will employ this Ricci integrability conditions on several occasions: For example, when we prove invariance of the normal sectional curvatures S ! 4 when we give an upper bound of the S ! 3 when we consider evolute type surfaces and the curvatures of their normal bundles. 1 Problem Statement In the same way as we derived the Riemannian curvature tensor from the Gauß integrability conditions we now proceed with deriving the curvature tensor of the normal bundle from the Ricci integrability conditions.

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