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Kenneth Wilkins
Kenneth Wilkins

Differential Geometry Of Curves And Surfaces ((BETTER))

Content: This will be an introduction to some of the ``classical'' theory of differential geometry, as illustrated by the geometry of curves and surfaces lying (mostly) in 3-dimensional space. The manner in which a curve can twist in 3-space is measured by two quantities: its curvature and torsion. The case a surface is rather more subtle. For example, we have two notions of curvature: the gaussian curvature and the mean curvature. The former describes the intrinsic geometry of the surface, whereas the latter describes how it bends in space. The gaussian curvature of a cone is zero, which is why we can make a cone out of a flat piece of paper. The gaussian curvature of a sphere is strictly positive, which is why planar maps of the earth's surface invariably distort distances. One can relate these geometric notions to topology, for example, via the so-called Gauss-Bonnet formula. This is mostly mathematics from the first half of the nineteenth century, seen from a more modern perspective. It eventually leads on to the very general theory of manifolds.

Differential Geometry of Curves and Surfaces

Aims: To gain an understanding of Frenet formulae for curves, the first and second fundamental forms of surfaces in 3-space, parallel transport of vectors and gaussian curvature. To apply this understanding in specific examples.

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss,[1] who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. The differential geometry of surfaces is concerned with a mathematical understanding of such phenomena. The study of this field, which was initiated in its modern form in the 1700s, has led to the development of higher-dimensional and abstract geometry, such as Riemannian geometry and general relativity.[original research?]

A surprising result of Carl Friedrich Gauss, known as the theorema egregium, showed that the Gaussian curvature of a surface, which by its definition has to do with how curves on the surface change directions in three dimensional space, can actually be measured by the lengths of curves lying on the surfaces together with the angles made when two curves on the surface intersect. Terminologically, this says that the Gaussian curvature can be calculated from the first fundamental form (also called metric tensor) of the surface. The second fundamental form, by contrast, is an object which encodes how lengths and angles of curves on the surface are distorted when the curves are pushed off of the surface.

Despite measuring different aspects of length and angle, the first and second fundamental forms are not independent from one another, and they satisfy certain constraints called the Gauss-Codazzi equations. A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface.

Using the first fundamental form, it is possible to define new objects on a regular surface. Geodesics are curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. They are very directly connected to the study of lengths of curves; a geodesic of sufficiently short length will always be the curve of shortest length on the surface which connects its two endpoints. Thus, geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface.

Each of these surfaces of constant curvature has a transitive Lie group of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's uniformization theorem (see below).

For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a Riemannian metric on the surface through line elements and area elements. Classically in the nineteenth and early twentieth centuries only surfaces embedded in R3 were considered and the metric was given as a 22 positive definite matrix varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a manifold, a topological space where the smooth structure is given by local charts on the manifold, exactly as the planet Earth is mapped by atlases today. Changes of coordinates between different charts of the same region are required to be smooth. Just as contour lines on real-life maps encode changes in elevation, taking into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each local chart. In each local chart a Riemannian metric is given by smoothly assigning a 22 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the Jacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensional Riemannian manifold.

Curves on a surface which minimize length between the endpoints are called geodesics; they are the shape that an elastic band stretched between the two points would take. Mathematically they are described using ordinary differential equations and the calculus of variations. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric are analytic.

Isothermal coordinates are known to exist in a neighbourhood of any point on the surface, although all proofs to date rely on non-trivial results on partial differential equations.[71] There is an elementary proof for minimal surfaces.[72]

The classical approach of Gauss to the differential geometry of surfaces was the standard elementary approach[90] which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century. The notion of connection, covariant derivative and parallel transport gave a more conceptual and uniform way of understanding curvature, which not only allowed generalisations to higher dimensional manifolds but also provided an important tool for defining new geometric invariants, called characteristic classes.[91] The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.[92]

As you can see, curves get short shrift. The essentials seem to be there, but do Carmo wants to move on and get to some surfaces. And there is lots of good stuff on surfaces! Essentially all of the theorems I learned in a first year Riemannian geometry course in graduate school makes an appearance here in the special case of a surface. (Mmmmm, turkey.)

Still, I wonder what i might have missed by skipping on learning the "classical" differential geometry. The book has a chapter about hypersurfaces but i'm still a bit worried that i might miss something important.

You won't miss much by dropping curves and surfaces: every important article I studied, browsed or heard about published in the last 60 years in differential geometry by such luminaries as Thom, Milnor, Atiyah, Hirzebruch, Perelman,...contains little or no reference to curves and surfaces.On the other hand if you spend your time on Codazzi equations, Frenet-Serret frames and umbilic points you might have no time left for principal bundles, Stiefel-Whitney or Chern classes, cobordism,etc. and that means you will have little chance of understanding anything in modern differential geometry.Of course it would be great to combine the mastery of both the exquisitely detailed classical results in one or two dimensions and the general powerful modern techniques of differential geometry/topology, but if you want to arrive at the frontier of research in a reasonable time you will have to favour the latter over the former.

I agree that you'll be fine if you go straight into manifolds before curves and surfaces. But it is simply NOT true (as stated in another answer) that surface is of little interest in modern different geometry, in particular in area related to PDE and analysis.

On the other hand, the use of surface theory is extremely essential in studying manifolds of positive curvatures, puesdo-holomorphic curves in symplectic manifolds etc, which are all active research directions in modern differential geometry.

Frenet-Seret theory of curves in the plane and in 3-space, examples; local theory of surfaces in 3-space: first and second fundamental forms, Gauss map and Gauss curvature, geodesics and parallel transport, theorema egregium, mean curvature and minimal surfaces. Prerequisites: One of MATH 102, 125 or 127 and one of MATH 209, 215 or 217. 041b061a72


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