By Tim Hoffmann

Discrete differential geometry investigates discrete analogs of items of
smooth differential geometry. hence, throughout the notes I seek advice from various
notions of classical differential geometry. yet whereas wisdom of uncomplicated dif-
ferential geometry is naturally precious, lots of the fabric could be under-
standable with out figuring out the sleek beginning of a few of the notions.

The fabric coated during this ebook is by means of nomeans a entire overview
of the rising box of discrete differential geometry yet i am hoping that it can

Similar differential geometry books

The topology of fibre bundles by Norman Steenrod PDF

Fibre bundles, now a vital part of differential geometry, also are of significant significance in glossy physics - equivalent to in gauge idea. This publication, a succinct advent to the topic by way of renown mathematician Norman Steenrod, was once the 1st to offer the topic systematically. It starts with a basic advent to bundles, together with such subject matters as differentiable manifolds and protecting areas.

Get Differential geometry and complex analysis: a volume PDF

Chavel I. , Farkas H. M. (eds. ) Differential geometry and intricate research (Springer, 1985)(ISBN 354013543X)(236s)

Theorems on regularity and singularity of energy minimizing by Leon Simon PDF

The purpose of those lecture notes is to offer an basically self-contained creation to the elemental regularity thought for power minimizing maps, together with fresh advancements in regards to the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of adaptations is no longer required.

Extra info for Discrete differential geometry

Sample text

What is left to show is that a∗ + b∗ + c∗ + d∗ = 0. Conjugating the left hand side and multiplying it with ac results in αc + β ac α α ac + αa + β = αc + β d + αa + β b = 0. b d β β This shows that we can integrate the dual edges into closing quadrilaterals. Remark. As in the smooth case dualizing is a duality: F ∗∗ = F up to translation. 2 Minimal surfaces are known to be isothermic and the dual surface is their Gauß map (it maps into the unit sphere and thus is a conformal map into S 2 ). In fact minimal surfaces can be characterized by this property and the dual of any conformal map into S 2 is minimal.

17 (and Lemma) Let F be s-isothermic with centres c and radii r. F ∗ given by c∗1 − c∗ = c1 − c , r1 r c∗2 − c∗ = − c2 − c r2 r 1 r is s-isothermic again and called a dual surface of F . r∗ = 63 64 Tim Hoﬀmann Lecture 14 Proof. we have to show, that F ∗ is well deﬁned and s-isothermic. If F is s-isothermic then its lift Fˆ solves a Moutard equation and we have 1 1 + =λ r r12 1 1 + r1 r 2 c c12 + =λ r r12 , c1 c2 + r1 r2 2 and |c|2 + r2 + |c12 |2 + r12 = λ (|c1 |2 + r12 + |c2 |2 + r22 ) which ﬁxes λ.

Proof. One can interpret the sequence γk = zk,0 as a discrete curve and γˆk = zk,1 as a μ-Darboux transform of it. Then starting with z˜0,0 we can create a λ-Darboux transform for γ. The Binachi permutability now states that γˆ˜ k = z˜k,1 is given uniquely such that γˆ˜ k = z˜k,1 is a λ-Darboux transform of γˆk = zk, 1 and a μ-Darboux transform of γ˜k . Iterating this deﬁnes the map z˜ uniquely and satisﬁes the stated cross-ratio conditions. 40 Lecture 8 Tim Hoﬀmann 41 Figure 12: A Tractrix and Darboux transform of the straight line.