N that assigns to a general point x 2x the kth osculating space tk x. Very short proof of the global gaussbonnet theorem. If we regard unit vectors as points on the unit sphere s2. A bijection is a map from xto y which is injective. We are measuring to which extent is the map from s to r3 given by p np called the gauss map di. It would be interesting to find an example of a closed set that is not weakly closed, that has nonempty interior and is such that the gauss map is not surjective. S, r3 is a smooth embedded oriented surface in r3, then the gauss map g. The elevation function on a smoothly embedded 2manifold in r3 re. Pn be an irreducible projective ndimensional variety over c, that is kjet spanned at the general point. This, together with the results of 1, shows that the characterisation of brillnoetherpetri curves with non surjective gauss wahl map as hyperplane sections of k3surfaces and limits thereof, obtained in 3, is optimal. Rank of the 2nd gaussian map for general curves 3 lemma 2. If youre good at picturing this, imagine travelling along a short path on a curved surface, and then travelling along the gauss maps image of that path. It is also trivially holomorphic, and thus, if nonconstant, its image is open.
We also say that the function is a surjection in this case. Cn is bijective if and only if the preimage of every complex line is connected and simplyconnected the proof uses the fact that the hopf map has no continuous sections. Gauss maps a surface in euclidean space r3 to the unit sphere s2. Computing elevation maxima by searching the gauss sphere. Finally in section 4 we will show how bertram, ein and lazarsfelds technique applied to general curves leads to a result on surjectivity of gaussian maps l. The notion of higher gauss maps here was introduced by fyodor l. A consists in showing that all the adic representations are surjective. The gauss map can be defined for hypersurfaces in r n as a map from a hypersurface to the unit sphere s n. Amod be additive functors such that gis left adjoint to. R3, choose a plane p perpendicular to v and very far.
A gauss map on s is a continuous function n that assigns to each point p. A function whose range is equal to its codomain is called an onto or surjective function. Theta divisors whose gauss map has a fiber of positive. Chapter 4 characters and gauss sums universiteit leiden. The area surrounding the point on the surface is thus mapped to an area on the unit sphere.
Rajeev january 27, 2009 1 statement of gausss law the electric. Because t is surjective, is ginvariant and must be haar measure. R to s be a surjective ring homomorphism and i be an ideal of r. The vector 000 is called the curvature vector at p. Given a set x, the power set 2x is the set of all subsets of x, including the empty set and xitself. We prove that c lies on a polarized k3 surface, or on a limit thereof, if and only if the gausswahl map for cis not surjective. Now consider example 3, the continued fraction example. If we regard unit vectors as points on the unit sphere s2, the we can think of n as a map n. Zak as a generalization of both ordinary gauss maps and. In this case a point on the submanifold is mapped to its oriented tangent subspace. Otherwise the set of parabolic points form a curve, the parabolic curve with equation h detf x ix j 0 the polynomial h is the hessian of f and is therefore a curve of degree 4dd 2. We have thus reproved that the gauss map is surjective for convex sets, as stated in the original question. This will lead us to the concept of second fundamental form, which is a quadratic form associated to s at the point p.
Optimization the process of nding the minimum or maximum value of an objective function e. A corollary of the gauss lemma university of michigan. This, together with the results of 1, shows that the characterisation of brillnoetherpetri curves with nonsurjective gausswahl map as hyperplane sections of k3surfaces and limits thereof, obtained in 3, is optimal. How do i visualize the gauss map and its derivative. If a geodesic is a plane curve, then it is a line of curvature. Let kbe a number eld and let akbe a principally polarized abelian variety of dimension d2. Describe the image of the gauss map for the surfaces. Motivation in gauss fundamental work on the geometry of surfaces in r3, he introduced an important object of study, the gauss map, that encodes how a surface stretches inside of r3. This is a continuous map from a compact space to a hausdorff space, and thus its image is closed.
A gauss map on s is a continuous function n that assigns to each point p 2s a unit normal vector np. If the gauss map is inseparable, all points are parabolic or planar. C where p is stereographic projection onto the wplane. Using gauss maps to detect intersections by frederico xavier.
The injectivity of the extended gauss map of general projections 33 apointx. Y such that given any x 2 x, there is a neighborhood u of x that is evenly covered by p, that is, p 1u can be represented as ivi, where vi is a collection of disjoint open sets of y. M are two surfaces and f is a surjective smooth map f. Projective varieties admitting an embedding with gauss map. So the gauss map is surjective onto the sphere just from the points of nonnegative curvature. In wl and in bm it is proved that if a curve lies on a k3 surface, then the gaussian map cannot be surjective. Group homomorphism from znz to zmz when m divides n.
The gauss map itself takes a surface, and maps each point onto a sphere at the point where the tangent planes are the same. A g, and noethers theorem says it is surjective if and only if xis nonhyperelliptic. Tangentplane that given a parameterization of a regular surface, we can define a field of unique surface normals the space of all possible normals lies in the unit sphere in, so we can identify with each normal its twodimensional location in the sphere. But the domain is inhabited and the codomain is connected, so this map must be surjective. The gauss map gpreserves the gauss measure, that is g. Moreover, if we write x maxspeca and y maxspecb, then the corresponding map y.
Sernesi abstract let c be a brillnoetherpetri curve of genus g 12. The starting idea of this paper is that, in view of 1. The maximum corank for a curve of genus g is 3 g 2, which is achieved by any hyperelliptic curve see w2 or. If a curve c is embedded in projective space by a very ample line bundle l, the gaussian map. The gauss map maps the unit normal of a surface on the right to the unit sphere on the left. We can recapture the bilinear form, and hence the map dn p itself, by polarizing the quadratic form q, and hence lose no information by focusing on q. The image of an ideal under a surjective ring homomorphism is. Recall that a polynomial is called monic if its leading coe cient is 1. The injectivity of the extended gauss map of general. On the surjectivity of the canonical gaussian map for. R, since the image of this function is the set of all real numbers.
Epstein, the hyperbolic gauss map and quasiconformal reflections 97. The map w 1 is called the wahl map, and it is related to important deformation and extendability properties of the canonical image of the curve cf. Elliptic curves with surjective adelic galois representations. Theorem 1 the gauss map is surjective if m is compact. Though not published till 2007, this paper ct has been highly in uential since its circulation as a preprint in 1984. From the proposition we see that the main work involved in showing that a given abelian surface ak has surjective adelic representation. S s2 is just the function g, with s2 the riemann sphere 2. Foranyoddg11,thereexistbrillnoetherpetricurveswhich are smooth hyperplane sections of a unique surface s. Math 501, homework 6 the gauss map and curvature due. Ranktwo vector bundles on halphen surfaces and the gauss. Nonlinear leastsquares problems with the gaussnewton. Now pull back the volume form of the sphere and use change of variables. Let v denote a two dimensional vector space over r. Mat 380, homework 3, due oct 15 there are 11 problems.
Most surfaces have two possible gauss maps, corresponding to the two. The maximum corank for a curve of genus g is 3 g 2, which is achieved by any hyperelliptic curve see w2 or cm2. Ways to prove the fundamental theorem of algebra mathoverflow. Let c be a curve of genus g 1 with general moduli, l. We prove that a map znz to zmz when m divides n is a surjective group homomorphism, and determine the kernel of this homomorphism.
R3 be a parameterization of cby arc length centered at p, i. The image of the higher gauss map for a projective variety is discussed. Pn corresponds to a 1dimensional vector subspace lxofv and one has ker. R n for a general oriented ksubmanifold of r n the gauss map can also be defined, and its target space is the oriented grassmannian, i. What can one say about the topology of points of zero gauss curvature on the surface say if it has measure zero. Chapter 3 characters and gauss sums universiteit leiden. The words mapping or just map are synonyms for function. The kernel of this map is a principal ideal gt in at, and gis monic. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the. Computing elevation maxima by searching the gauss sphere bei wang 1, herbert edelsbrunner,2, and dmitriy morozov 1 department of computer science, duke university, durham, north carolina, usa 2 geomagic, research triangle park, north carolina, usa abstract.
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