Category: Complex geometry of real hyperboloids

Complex geometry of real hyperboloids

Abstract: In this paper, we study the local Cauchy-Riemann embeddability of strictly pseudoconvex real hyperboloids into spheres. By solving a CR analogue of the Gauss equation, we prove that is CR-embeddable into spheres with a CR co-dimension if and only if it is spherical.

References [Enhancements On Off] What's this? Chern and J. MoserReal hypersurfaces in complex manifoldsActa Math. Peter EbenfeltXiaojun Huangand Dmitri ZaitsevThe equivalence problem and rigidity for hypersurfaces embedded into hyperquadricsAmer.

MR 4. James J. Faran VThe nonimbeddability of real hypersurfaces in spheresProc. Xiaojun HuangOn a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensionsJ.

complex geometry of real hyperboloids

Differential Geom. MR 7. WebsterPseudo-Hermitian structures on a real hypersurfaceJ. Differential Geometry 13no. MR WebsterSegre polar correspondence and double valued reflection for general ellipsoidsAnalysis and geometry in several complex variables Katata, Trends Math.

Special issue for S. Chern, S. MR 2. Ebenfelt, P. MR 3. Faran, J.

complex geometry of real hyperboloids

MR 5. Forstneric, F. MR 6. Huang, X. MR 8. Lempert, L. MR 9. Webster, S.Consider the two parallel planes from the pencils of planes of which the given parallel lines are the axes. Hence the two parallel lines must also intersect on this line.

Similarly, using spectral decomposition and row scaling, the metric diagonalization of a general paraboloid takes the form. Finally, using permutation matrices and row scaling, the metric diagonalization of a two-sheeted hyperboloid takes the form.

Inversely, this means if two matrices have different matrix signatures then they can not be congruent. We know that the matrix signature of an ellipsoid is 2 and that of a one-sheeted hyperboloid is 0. Hence these matrices can not be congruent.

As ellipsoids, paraboloids and two-sheeted hyperboloids have the same matrix signature of 2, they are congruent. The only rotation matrix that has a repeated eigenvalue of 1 is the identity matrix, as the eigenvectors of such a matrix must span the whole vector space and the identity matrix is the only one that does so.

Hence the eigenvectors corresponding to the real-valued eigenvalue will be coincident. We also know that rotations preserve circular points at infinity. Toggle navigation Immensely Happy. Show that a real projective transformation of 3-space can map an ellipsoid to a paraboloid or hyperboloid of two sheets, but cannot map an ellipsoid to a hyperboloid of one sheet i. What do the eigenvectors corresponding to the complex eigenvalues represent?

References Wikipedia. Section 3, Notes on Rotations.Hyperboloidthe open surface generated by revolving a hyperbola about either of its axes. Revolution of the hyperbola about its conjugate axis generates a surface of one sheet, an hourglass-like shape see figureleftfor which the second term of the above equation is positive.

The intersections of the surface with planes parallel to the xz and yz planes are hyperbolas. Intersections with planes parallel to the xy plane are circles or ellipses. Revolution of the hyperbola about its transverse axis generates a surface of two sheets, two separate surfaces see figure, rightfor which the second term of the general equation is negative. Intersections of the surface s with planes parallel to the xy and xz planes produce hyperbolas. Cutting planes parallel to the yz plane and at a distance greater than the absolute value of aafrom the origin produce circles or ellipses of intersection, respectively, as a equals b or a is not equal to b.

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Learn More in these related Britannica articles: hyperbola.

Hyperboloid structure

Hyperbolatwo-branched open curve, a conic section, produced by the intersection of a circular cone and a plane that cuts both nappes see cone of the cone. As a plane curve it may be defined as the path locus of a point moving so that the ratio of the distance…. History at your fingertips. Sign up here to see what happened On This Dayevery day in your inbox! Email address. By signing up, you agree to our Privacy Notice.

Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox.In geometrya hyperboloid of revolutionsometimes called a circular hyperboloidis the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalingsor more generally, of an affine transformation.

A hyperboloid is a quadric surfacethat is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinderhaving a center of symmetryand intersecting many planes into hyperbolas.

A hyperboloid has three pairwise perpendicular axes of symmetryand three pairwise perpendicular planes of symmetry. Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are the axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations:. Both surfaces are asymptotic to the cone of the equation.

There are two kinds of hyperboloids.

Rotating Hyperboloids

It is a connected surfacewhich has a negative Gaussian curvature at every point. This implies near every point the intersection of the hyperboloid and its tangent plane at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are lines and thus the one-sheet hyperboloid is a doubly ruled surface. The surface has two connected components and a positive Gaussian curvature at every point.

Thus the surface is convex in the sense that the tangent plane at every point intersects the surface only in this point. This property is called Wren 's theorem.

complex geometry of real hyperboloids

A hyperboloid of one sheet is projectively equivalent to a hyperbolic paraboloid. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too. Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case see circular section. The hyperboloid of two sheets does not contain lines.

The discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation. Obviously, any two-sheet hyperboloid of revolution contains circles.

Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry. More generally, an arbitrarily oriented hyperboloid, centered at vis defined by the equation. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue.

The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues. Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a pseudo-Euclidean space one has the use of a quadratic form :.In mathematicscomplex geometry is the study of complex manifoldscomplex algebraic varietiesand functions of several complex variables.

Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Broadly, complex geometry is concerned with spaces and geometric objects which are modelled, in some sense, on the complex plane. Features of the complex plane and complex analysis of a single variable, such as an intrinsic notion of orientability that is, being able to consistently rotate 90 degrees counterclockwise at every point in the complex planeand the rigidity of holomorphic functions that is, the existence of a single complex derivative implies complex differentiability to all orders are seen to manifest in all forms of the study of complex geometry.

As an example, every complex manifold is canonically orientable, and a form of Liouville's theorem holds on compact complex manifolds or projective complex algebraic varieties. Complex geometry is different in flavour to what might be called real geometry, the study of spaces based around the geometric and analytical properties of the real number line.

For example, whereas smooth manifolds admit partitions of unitycollections of smooth functions which can be identically equal to one on some open setand identically zero elsewhere, complex manifolds admit no such collections of holomorphic functions. Indeed, this is the manifestation of the identity theorema typical result in complex analysis of a single variable. In some sense, the novelty of complex geometry may be traced back to this fundamental observation.

It is true that every complex manifold is in particular a real smooth manifold. However, complex geometry is not typically seen as a particular sub-field of differential geometrythe study of smooth manifolds. In particular, Serre 's GAGA theorem says that every projective analytic variety is actually an algebraic varietyand the study of holomorphic data on an analytic variety is equivalent to the study of algebraic data.

This equivalence indicates that complex geometry is in some sense closer to algebraic geometry than to differential geometry. Another example of this which links back to the nature of the complex plane is that, in complex analysis of a single variable, singularities of meromorphic functions are readily describable.

In contrast, the possible singular behaviour of a continuous real-valued function is much more difficult to characterise. As a result of this, one can readily study singular spaces in complex geometry, such as singular complex analytic varieties or singular complex algebraic varieties, whereas in differential geometry the study of singular spaces is often avoided. In practice, complex geometry sits in the intersection of differential geometry, algebraic geometry, and analysis in several complex variablesand a complex geometer uses tools from all three fields to study complex spaces.

Typical directions of interest in complex geometry involve classification of complex spaces, the study of holomorphic objects attached to them such as holomorphic vector bundles and coherent sheavesand the intimate relationships between complex geometric objects and other areas of mathematics and physics.

Complex geometry is concerned with the study of complex manifoldsand complex algebraic and complex analytic varieties. Here the definitions of and relations between these types of spaces are presented.

In contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. In order to define a general complex algebraic or complex analytic variety, one requires the notion of a locally ringed space. Again we also by convention require this locally ringed space to be irreducible.

We say a complex variety is smooth or non-singular if it's singular locus is empty. That is, if it is equal to its non-singular locus.

The Geometry of Antoni Gaudi

By the implicit function theorem for holomorphic functions, every complex manifold is in particular a non-singular complex analytic variety, but is not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a projective complex algebraic variety. When a complex variety is non-singular, it is a complex manifold. More generally, the non-singular locus of any complex variety is a complex manifold.

Complex manifolds may be studied from the perspective of differential geometry, whereby they are equipped with extra geometric structures such as a Riemannian metric or symplectic form.

In order for this extra structure to be relevant to complex geometry, one should ask for it to be compatible with the complex structure in a suitable sense. Serre's GAGA theorem asserts that projective complex analytic varieties are actually algebraic. Whilst this is not strictly true for affine varieties, there is a class of complex manifolds that act very much like affine complex algebraic varieties, called Stein manifolds. Another way in which Stein manifolds are similar to affine complex algebraic varieties is that Cartan's theorems A and B hold for Stein manifolds.

Examples of Stein manifolds include non-compact Riemann surfaces and non-singular affine complex algebraic varieties. Calabi-Yau manifolds have found use in string theory and mirror symmetrywhere they are used to model the extra 6 dimensions of spacetime in dimensional models of string theory. Examples of Calabi-Yau manifolds are given by elliptic curvesK3 surfaces, and complex Abelian varieties.Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet.

Often these are tall structures such as towers where the hyperboloid geometry's structural strength is used to support an object high off the ground, but hyperboloid geometry is also often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov — Hyperbolic structures have a negative Gaussian curvaturemeaning they curve inward rather than curving outward or being straight.

As doubly ruled surfacesthey can be made with a lattice of straight beams, hence are easier to build than curved surfaces that do not have a ruling and must instead be built with curved beams.

Hyperboloid structures are superior in stability towards outside forces compared with "straight" buildings, but have shapes often creating large amounts of unusable volume low space efficiency and therefore are more commonly used in purpose-driven structures, such as water towers to support a large masscooling towers, and aesthetic features. With cooling towersa hyperbolic structure is preferred.


At the bottom, the widening of the tower provides a large area for installation of fill to promote thin film evaporative cooling of the circulated water. As the water first evaporates and rises, the narrowing effect helps accelerate the laminar flowand then as it widens out, contact between the heated air and atmospheric air supports turbulent mixing.

In the s, Shukhov began to work on the problem of the design of roof systems to use a minimum of materials, time and labor. His calculations were most likely derived from mathematician Pafnuty Chebyshev 's work on the theory of best approximations of functions.

Shukhov's mathematical explorations of efficient roof structures led to his invention of a new system that was innovative both structurally and spatially. By applying his analytical skills to the doubly curved surfaces Nikolai Lobachevsky named "hyperbolic", Shukhov derived a family of equations that led to new structural and constructional systems, known as hyperboloids of revolution and hyperbolic paraboloids. The steel gridshells of the exhibition pavilions of the All-Russian Industrial and Handicrafts Exposition in Nizhny Novgorod were the first publicly prominent examples of Shukhov's new system.

Two pavilions of this type were built for the Nizhni Novgorod exposition, one oval in plan and one circular. The roofs of these pavilions were doubly curved gridshells formed entirely of a lattice of straight angle-iron and flat iron bars.

Shukhov himself called them azhurnaia bashnia "lace tower", i. The patent of this system, for which Shukhov applied inwas awarded in Shukhov also turned his attention to the development of an efficient and easily constructed structural system gridshell for a tower carrying a large gravity load at the top — the problem of the water tower.

His solution was inspired by observing the action of a woven basket holding up a heavy weight. Again, it took the form of a doubly curved surface constructed of a light network of straight iron bars and angle-iron. Over the next twenty years, he designed and built close to two hundred of these towers, no two exactly alike, most with heights in the range of 12m to 68m.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. The following youtube animation which rotates the embedded cross cap presumably in the xw plane suggests line AB is actually a circle.

Therefore what looks like a pinch point B and double line AB in the 3D projection of the cross cap is actually a circle partially pointing in 4D. However, because the animation have not explored other rotations, it is hard to deduce the nature of the pinch point A. Using the answer from this linkline AB corresponds to the Whiteney's umbrella region of the cross cap, which when understood in a 4D perspective, the "unusual line intersection" can be reasoned as a parabola orienting in the yw plane and hence there is no actual intersection in 4D.

However, as the gluing continues, since it is required to glue opposite edges of the whiteney's umbrella together, it seems there is no way to prevent the aforementioend yw parabola to become a yw circle and hence suggesting the two surfaces do touch at the origin pinch point A. I then suspect the actual geometry of the pinch point A, if the neighbourhood is rotated slightly and the result projected back to 3D will be one of the following:.

The left scenario will suggest even if the surface don't intersect itself in 4D, it still need to touch at the origin to complete the gluing process, thus suggesting A is indeed some kind of double point.

Therefore the region near the origin would trace out an actual figure 8 loop. The middle scenario is perhaps the most interesting, as having the twist right at the middle, it suggests the origin is possibly a saddle point. The figure 8 loop seen in the cross cap 3D projection can then be explained as an artifact due to the projection process mapping distinct points along the w axis to the same point in xyz. But with the lack of experience on how to probe this from the embedding equations because the form of the equations there suggest I am dealing with some kind of hyperboloids, and it is unclear how I can mathematically take a section of the neighbourhood near the origin or even the circle-line ABI am unsure of the geometry near the origin, hence the actual nature of the pinch point A.

Main: What is the nature of the pinch point A, is it really some kind of double point even in 4D? Math approach: What direction should I try in order to choose a good section from the embedding equations to mathematically investigate the region near the origin and circle-line AB in order to learn how the surface twists there in 4D? Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Real projective plane: Geometry of the double line and the centre pinch point Ask Question. Asked 3 years, 6 months ago. Active 3 years, 6 months ago. Viewed times. I then suspect the actual geometry of the pinch point A, if the neighbourhood is rotated slightly and the result projected back to 3D will be one of the following: The left scenario will suggest even if the surface don't intersect itself in 4D, it still need to touch at the origin to complete the gluing process, thus suggesting A is indeed some kind of double point.

Secret Secret 2, 12 12 silver badges 23 23 bronze badges. As written, it is not even a map of the projective plane.

Hwang Jan 9 '17 at Active Oldest Votes.

complex geometry of real hyperboloids

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