By Tim Hoffmann

Discrete differential geometry investigates discrete analogs of items of

smooth differential geometry. hence, throughout the notes I check with various

notions of classical differential geometry. yet whereas wisdom of uncomplicated dif-

ferential geometry is naturally necessary, many of the fabric may be under-

standable with out realizing the sleek foundation of some of the notions.

The fabric coated during this publication is by way of nomeans a entire overview

of the rising box of discrete differential geometry yet i'm hoping that it can

serve as an creation.

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**Additional resources for Discrete differential geometry**

**Example text**

Cr(γ, γ1 , γ˜1 , γ˜ ) = l12 . For the absolute value of the cross-ratio this is obvious. For its argument, observe that the triangles (γ, γ1 , γ˜ ) and (˜ γ , γ1 , γ˜1 ) are similar giving that the arguments of (γ − γ1 )/(γ1 − γ˜1 ) and (˜ γ1 − γ˜ )/(˜ γ − γ) sum to a multiple of 2π. So indeed the euclidean Darboux transform is a special case of the one we formulated for the P1 picture. ❈ • One can show that the Darboux transformation commutes with the tangential and (m)KdV ﬂows: If γ and a Darboux transform γ˜ evolve with one of these ﬂows they stay related by a Darboux transformation for all times.

Representation of ❍ in the complex 2 × 2-matrices There is a M at(2, ❈) via 1∼ = 1 0 0 1 , ✐∼ = −iσ1 = −i 0 1 1 0 42 , ❥∼ = −iσ2 = −i gl(2, ❈) = 0 −i i 0 , Lecture 8 Tim Hoﬀmann 1 0 0 −1 ❦∼ = −iσ3 = −i 43 . The matrices σk are sometimes called Pauli spin matrices. In analogy to the complex numbers one deﬁnes for q ∈ ❍, q = α + β ✐ + γ ❥ + δ ❦ • Re(q) = α • Im(q) = q − Re(q) = β ✐ + γ ❥ + δ ❦ • q¯ = q − 2Im(q) = α − β ✐ − γ ❥ − δ ❦ √ • |q| = q q¯ = α2 + β 2 + γ 2 + δ 2 and ﬁnds q −1 = q¯ . |q|2 Note that in contrast to the complex case Im(q) is not a real number (unless it is 0) but lies in the imaginary quaternions Im❍ := span{✐, ❥, ❦}.

S-isothermic maps are build from spheres (which are represented by points in deSitter space) that have for each quadrilateral a common orthogonal circle or a common pair of points: The four spheres of a quadrilateral are collinear, so they span a 3-space. The orthogonal compliment of that 3-space is a 2dimensional subspace, that – if space-like – describes a circle (the 1-dim set of space-like unit vectors in the 2-space give all the spheres that contain the circle), but if time-like contains exactly two light-like directions that give two points contained in all four spheres (remember: a point lies on a sphere iﬀ their vectors in Minkowski ❘5 are perpendicular).