# Download Geometry Symposium Utrecht 1980: Proceedings of a Symposium by P. Baird, J. Eells (auth.), E. Looijenga, D. Siersma, F. PDF

By P. Baird, J. Eells (auth.), E. Looijenga, D. Siersma, F. Takens (eds.)

**Read Online or Download Geometry Symposium Utrecht 1980: Proceedings of a Symposium Held at the University of Utrecht, The Netherlands, August 27–29, 1980 PDF**

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**Extra resources for Geometry Symposium Utrecht 1980: Proceedings of a Symposium Held at the University of Utrecht, The Netherlands, August 27–29, 1980**

**Example text**

Let X : M2 > ~4 be a smooth immersion of a surface without boundary, not necessarily orientable, into vectors given by ~*X : M2 V in ~(Z) for 4-space. e. the composition will have a finite number of singularities where the rank of not 2 . These singularities occur at points lies in the tangent plane to X(M 2) at X(u) . In order for X , we require further that for each singular point of a small disc about u in M2 u u of M2 such that V to be general the image under ~oX will be topologically a cone over a figure eight, a singularity known as a Whitney pinch point or an umbrella point.

The results of Section conditions | and 2 in order to formu- for the existence of a minimal hypersurfa- 52 I. The Ricci C o n d i t i o n and its Generalizations. We start by d i s c u s s i n g the classical result of Ricci for minimal surfaces in E 3 . I. Let any point of M M where the Gauss curvature i) the metric is negative, has constant curvature ds g = #-K ds 2 iii) the curvature K ds 2 . T h e n at the following three con- K m | ; has constant curvature Km 0 ; satisfies (I) A log(-K) = 4K , where Proof K En : d~ 2 = - Kds 2 ii) the metric .

I 13 J (uij) = ,e) (u B) , ,e i) spaces : e' = l ~ u~seB are orthogonal matrices, m~l~ = (del'e~) = Z uiju 8wjB ~! oJ. m. i i~ = Z are called the second fundamental forms. ~. lJ~ i J h.. If then ll(v) = Z v II is the second fundamental form in the direction v ; the second fundamental form is 65 thus a normal-valued quadratic differential form. , (23a) Z h.. = 0 = if 0 . equations dm. = Z ~. A ~.. z j JZ which shows that Civita (mij) connection, (5) in three sets v . if of the other structure (24) in the normal vector IIfX i The pull-back , defines , the torsionless in the tangent in (4) gives bundle.