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.)
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Additional resources for Geometry Symposium Utrecht 1980: Proceedings of a Symposium Held at the University of Utrecht, The Netherlands, August 27–29, 1980
Example text
Plane so that during for w h i c h the compression the tangential We can choose the there will be only finitely many positions developable such points will correspond One way is to com- of X becomes tangent to t to critical points of non-degenerate and we may analyze each such non-general position Y . In general height functions + y H (Xt,) to show that the sum remains unchanged. A A tI+ = 1 + -At]I = -i A tH + = 0 t-E t t+c I At(2C ) = 0 At I+ = i A - + t~I = -i AH+=O t t-c t Another non-general the secant vector with Y situation occur if there is a double is in an asymptotic a ruling in the tangential T+(X) t+e developable directiononY.
Remark 2 : By considering an orientation on one of the polygons, say X , we may obtain slightly refined versions of the theorem in the case where the vertices X are in general position. We define I+(X,Y) to be the number of opposite double supports of the form x xi 1 but not V J Xi- I Xi+ I Xi+ 1 Xi_ 1 of side 32 + and ]I (X,Y) similarly is the number of same side double X. Y. i ] supports X. Y. -< t O X(u) = Y(v) which is not a transversal that X(u) - Y ( v ) # O and X(u) - Y(v) Y(v) but one of these points arguments similar to the above.
An obvious remark is that in order for a metric to be realizable on a minimal submanifold of En By Nash's Theorem, it must first of all be realizable on some submanifold of En that is no restriction if one allows the codimension to be large. But for small codimension it is a severe restriction. For example, for codimension one and dimension at least three, one cannot have at any point all sectional curvatures negative (by equation (25a) in w since each pair of ki,k j would have to have opposite signs.