differential geometry

0.1 Classical differential geometry

Differential geometry studies geometrical objects using techniques ofcalculus. In fact, its early history is indistiguishable from that ofcalculus — it is a matter of personal taste whether one chooses toregard Fermat’s method of drawing tangents and finding extrema as acontribution to calculus or differential geometry; the pioneering workof Barrow and Newton on calculus was presented in a geometricallanguage; Halley’s 1696 paper in which he announces his discovery that $\\displaystyle\\int\\frac{dx}{x}=\\log x+C$ is entitled quadrature of the hyperbola.

It is only later on, when calculus became more algebraic in outlookthat one can begin to make a meaningful separation between thesubjects of calculus and differential geometry. Early differentialgeometers studied such properties of curves and surfaces such as:computing their lengths and areas, finding tangents, constructingevolute, involute, and pedal curves, studying curvature and osculatingcircles, and finding envelopes and orthogonal curves to a given familyof curves. Of the various objects they studied, the cycloid deservesspecial mention. Originally discovered by Galileo, it seems to havebeen studied by just about every seventeenth century mathematician,much more than any other curve. Also, it is worth mentioning that thedeep connections between differential geometry and mechanics whichplay a prominent role in contemporary theoretical physics also havetheir origin at this time — for instance, the fact that the problemof geodesics on surfaces is related to the acceleration of particleswas already to known to the Bernoullis, even though the significanceof this similarity was not fully appreciated until Einstein.

0.2 Intrinsic geometry

A major turning point in differential geometry is marked by theappearance of Gauss’ memoir “Disquisitiones generales circasuperficies curvas”. In this memoir, Gauss first proposes theintrinsic point of view in geometry. Since his idea of has intrinsicgeometry has completely changed the outlook of geometry, it might notbe inappropriate to spend some time discussing it.

Ordinarily, when we think of a surface such as a cylinder or a sphere,we concieve of a locus in space ( $\\mathbb{R}^{3}$ ). However, imaginethat, like blind ants crawling on a sheet, we were confined to asurface and had no direct knowledge of the space in which the surfacewas situated. What could we conclude about the surface on which welive? Obviously, such constructions as normals to the surface orosculating spheres which are situated in the ambient space would bemeaningless to us. However, we would be still able to speak of thelengths of curves drawn on the surface, the angle between curves, andareas of portions of our surface since these can be defined in termsof measurements which are made on the surface without recourse to theambient space.

Considering differential geometry from this point of view, one comesto several interesting conclusions. One is that certain surfaces,such as a portion of a plane and a portion of a cylinder areindistinguishable from the intrinsic point of view. This result canbe something of a surprise because one is usually accostomed tothinking of planes and cylinders as rather different sorts of objects— planes are flat while cylinders are curved.

By itself this discovery is interesting, but perhaps not enough tofire a revolution in geometry. Hearing of it, one might come to theconclusion that intrinsic measurements alone are not sufficient todescribe the geometry of a surface and, hence, the subject ofintrinsic geometry is uninteresting. Further study shows that this isnot the case. While one may not be able to distinguish a plane from acylinder on the basis of intrinsic measurements alone, it is possibleto distinguish a portion of plane from a portion of sphere solely onthe basis of intrinsic measurements. Even more, it is possible todistinguish portions of spheres of different radii intrinsically.This, of course, is of interest not only because it shows that thestudy of intrinsic geometry is non-trivial, but because a quantitysuch as the radius of a sphere which is defined as the distance from apoint on the sphere to a point not on the surface (namely, the centreof the sphere) can, in fact, be deduced solely from measurements onthe surface of the sphere.

To prove such results, Gauss used the concept of curvature. Since theidea of curvature, in one from or another, plays an important role indifferential geometry to this day, let us say a few words about thiskey concept. The concept of curvature was developed in the eighteenthcentury as a measure how much a given curve or a surface deviates frombeing a straight line or a plane. In the case of a curve, thecurvature may be defined as the second derivative of the normal anglewith respect to arclength. In the case of a surface, the situation isa little more complicated — to describe the direction of the normalvector, one needs two angles instead of one and one can choose tocompute their directional derivatives along any direction tangent tothe surface. Because of this, one obtains a $2\\times 2$ matrix ofpartial derivatives instead of a single number. If one changes thecoordinates used to describe the ambient space, then the components ofthis matrix will undergo a linear transformation. To obtain aquantity which does not depend on an arbitrary choice of coordinatesand hence can be seen as describing geometric properties of thesurface, one should consider the eigenvalues of this matrix. Theseeigenvalues are known as the principal curvatures of the surface.

The remarkable theorem which Gauss proved was that, whilst theprincipal curvatures cannot be determined from intrinsic measurementsalone, their product can. This result, for instance, can be used toexplain the facts we mentioned about planes, cylinders, and spheres.For a plane, the two principal curvatures equal zero. For a cylinder,one principal curvature is zero and the other is positive. For asphere, both principal curvatures are equal and positive. Since theproduct equals zero for both the plane and the cylinder it isplausible that they are indistinguishable intrinsically. Since it isnot zero for the sphere, it follows from Gauss’ theorem that theintrinsic geometry of the sphere could not possibly be the same asthat of the plane and cylinder.

0.3 The concept of a manifold

The discovery of intrinsic geometry led thoughtful geometers such asRiemann (who was a student of Gauss), Clifford, and Mach to theconclusion that a “right and natural” approach to geometry shouldregard surfaces as geometrical spaces in their own right on a par withEuclidean and projective space. In the terminology of the mediaevalscholastic philosophers, one may say that they regarded the intrinsicproperties of surfaces as “essential” and extrinsic properties as“accidental”. To properly develop geometry from such a viewpoint,they needed to start with a definition of surface which made noreference to any sort of ambient space. Their quest for such adefinition led to the concept of a manifold.

The genesis of this concept can be seen as a diasappearing act akin tothat of the Chesire cat — just as the grin is all that remains ofthe cat when the cat disappears, so too a manifold is what remainswhen one starts with a parameterized surface and the space in whichthe surface is situated disappears.

A parameterized surface may be decribed as a suitable subset $S\\in\\mathbb{R}^{3}$ together with a smooth bijective map from an open subsetof $\\mathbb{R}^{2}$ to $S$ (which is called a parameterization of thesurface). Moreover, (pay careful attention because this will turn outto be the key fact that makes the concept of a manifold possible) onecan describe the same surface using many different parameterizations.Given two parameterizations $\\phi\\colon D_{1}\\subset\\mathbb{R}^{2}\\to S$ and $\\psi\\colon D_{2}\\subset\\mathbb{R}^{2}\\to S$ of the sameportion of surface, the two will be related by reparametrization; thatis to say there exists a smooth bijection $g_{\\phi\\psi}\\colon D_{2}\\to D_{1}$ such that $\\psi=\\phi\\circ g_{\\phi\\psi}$ .

By contrast, a manifold may be understood as a “parameterized set”.That is to say, a $n$ -dimensional manifold $M$ is a set together witha set of bijective maps from open sets of $\\mathbb{R}^{n}$ to $M$ (whichare called coordiante maps). As in the case of parameterizedsurfaces, a coordinate map need not have the whole of $M$ as itsrange. Moreover, we require that if two coordinate maps $\\phi\\colon D_{1}\\subset\\mathbb{R}^{2}\\to M$ and $\\psi\\colon D_{2}\\subset\\mathbb{R}^{2}\\to M$ describe the same subset of the manifold, thenthere exists a smooth bijective map $g_{\\phi\\psi}\\colon D_{2}\\to D_{1}$ such that $\\psi=\\phi\\circ g_{\\phi\\psi}$ . (Such a map is known asa coordinate transformation map or a transition function.) To makethis definition completely correct, we require a few more technicalassumptions but, in keeping with the spirit of this expsition, weshall not discuss them here and instead refer the reader to the entrynotes on the classical definition of a manifold for a carefuldefinition which takes technicalities into account.

As is obvious from the definition, if we make the further assumptionthat the objects of our set be points of Euclidean space and that thecorrespondence between points and pairs of numbers be continuous, werecover the definition of parameterized surface. However, we chooseto refrain from making any asumption about the nature of the elementsof our set. This freedom is exactly what allows us to regard twosurfaces with the same intrinsic geometry as the same manifold — toobtain one surface, we specify one mapping of our set into Euclideanspace one way and to obtain the other surface we specify a differentmapping.

Objects other than surfaces can be manifolds. Most obviously, theEuclidean plane itself is a manifold since its points can be describedby pairs of real numbers according to various coordiante systems(Cartesian coordinates, polar coordinates, etc.) Thus, the concept ofmanifold fulfills the desire of its inventors that Euclidean space andsurfaces be of the same ontological status.

Less obviously and more curiously, the set of functions which satisfythe differential equation

f^{\\prime\\prime}(t)+2f^{\\prime}(t)+3f(t)=0

is also a two-dimensional manifold! The reason is that we can specifya solution of this equation uniquely by giving the values of $f$ and $f^{\\prime}$ at a particular value of $t$ and, by the theorem on continuousdependence on initial conditions, $f(t_{1})$ and $f^{\\prime}(t_{1})$ can beexpressed as a continuous functions of $f(t_{2})$ and $f^{\\prime}(t_{2})$ for anytwo numbers $t_{1},t_{2}\\in\\mathbb{R}$ .

Another example of a manifold is the group of affine transforms of theline. Recall (or look back at the section on affine geometry above!)that an affine transformation is of the form $x\\mapsto ax+b$ .Hence, such a transform is specified by giving the two real numbers $a$ and $b$ . A group like this one which also happens to be amanifold is called a Lie group. The study of Lie groups forms animportant branch of group theory and is of relevance to other branchesof mathematics.

Because of examples like the two just exhibited, it has been possibleto apply the techniques of differential geometry in some ratherunlikely settings. Once geometric notions like tangent spaces andcurvature have been defined for manifolds (we shall indicate how thisis done in the next section) then one can speak of such things as thetangent space to the set of solutions of a differential equation orthe curvature of a group. While this may sound like a parlor stunt todemonstrate the generality of our definitions, it is more than that.By applying the techniques of differential geometry in such unlikelysettings, mathematicians have been able to win insights and proveresults about differential equations, groups and other mathematicalobjects which otherwise seemed intractable.

Before moving on to the next section, it might be worth pointing outthat, when speaking of manifolds, it is customary to refer to theelements of the set as points and the real numbers which label them aspoints. The reader should keep in mind that this is merely customaryterminology which derives from thinking of manifolds as ageneralization of parameterized surfaces and should no way beunderstood as suggesting that the elements of the set which compriseour manifold resemble points of a surface. As the examples show, theymay be functions, transformations, or other mathematical objects.

0.4 Structures on manifolds

In order to discuss such geometric notions as angles and lengths andperform interesting geometric constructions, one needs to equip one’smanifold with suitable structures. In classical differentialgeometry, these structures were provided by the ambient space, but nowthat the ambient space has disappeared, they must put in by hand.

In placing structures on a manifold, the notion of reparameterizationor change of coordinates which was built into the definition ofmanifold plays a crucial role. To explain how the process of imposingstructures proceeds, let us start with a simple example — tangentvectors on a manifold. In classical differential geometry, a tangentvector ${\\bf v}$ to a surface $S$ at a point ${\\bf p}\\in S$ is simplya vector in Euclidean space whose direction happens to be tangent tothe surface at the point ${\\bf p}$ . We could represent it graphicallyan arrow with its tail at ${\\bf p}$ which points along a tangent tothe surface.

Of course, this description won’t do if we don’t have an ambient spacein which to draw our arrow. Therefore, we need to look for adifferent description of our tangent vector. One possibility is toconsider a description of the vector in terms of its components.However, to define the components of a vector, one needs a basis. Ifour surface is specified parametrically, there is a natural choice ofbasis vectors, namely

\\begin{pmatrix}\\displaystyle\\frac{\\partial x}{\\partial s}&\\displaystyle\\frac{%\\partial y}{\\partial s}&\\displaystyle\\frac{\\partial z}{\\partial s}\\end{pmatrix}

and

\\begin{pmatrix}\\displaystyle\\frac{\\partial x}{\\partial t}&\\displaystyle\\frac{%\\partial y}{\\partial t}&\\displaystyle\\frac{\\partial z}{\\partial t}\\end{pmatrix}.

Of course, if we choose a different parameterization, we obtain adifferent pair of basis vectors. However, we can express one pair interms of the other pair using the chain rule. Thus, if a tangentvector ${\\bf v}$ has components $\\begin{pmatrix}v_{1}&v_{2}\\end{pmatrix}$ with respect to the basis corresponding to theparameterization in terms of parameters $s$ and $t$ , it will havecomponents $\\begin{pmatrix}{v^{\\prime}}_{1}&{v^{\\prime}}_{2}\\end{pmatrix}$ withrespect to the basis corresponding to the parameterization in termsof parameters $s^{\\prime}$ and $t^{\\prime}$ , where

\\begin{pmatrix}{v^{\\prime}}_{1}&{v^{\\prime}}_{2}\\end{pmatrix}=\\begin{pmatrix}v%_{1}&v_{2}\\end{pmatrix}\\begin{pmatrix}\\displaystyle\\frac{\\partial s}{\\partial s%^{\\prime}}&\\displaystyle\\frac{\\partial s}{\\partial t^{\\prime}}\\\\&\\\\\\displaystyle\\frac{\\partial t}{\\partial s^{\\prime}}&\\displaystyle\\frac{%\\partial t}{\\partial t^{\\prime}}\\end{pmatrix}.

These observations form the foundation of the definition of a tangentvector to a manifold. To define a tangent vector ${\\bf v}$ to atwo-dimensional manifold $M$ at a point ${\\bf p}$ , we shall associatea pair of numbers $\\begin{pmatrix}v_{1}&v_{2}\\end{pmatrix}$ to everycoordinate system which describes ${\\bf p}$ . The only restriction weimpose is that, given two coordinate systems, the pairs of numbers tobe associated to these systems be related by the transform

\\begin{pmatrix}{v^{\\prime}}_{1}&{v^{\\prime}}_{2}\\end{pmatrix}=\\begin{pmatrix}v%_{1}&v_{2}\\end{pmatrix}\\begin{pmatrix}\\displaystyle\\frac{\\partial s}{\\partial s%^{\\prime}}&\\displaystyle\\frac{\\partial s}{\\partial t^{\\prime}}\\\\&\\\\\\displaystyle\\frac{\\partial t}{\\partial s^{\\prime}}&\\displaystyle\\frac{%\\partial t}{\\partial t^{\\prime}}\\end{pmatrix}.

To define other structures on the manifold, we can follow a similar recipe:

1.
Identify what sort of mathematical object will represent thestructure in a coordinate system. In the case of a tangent vector,this was an $n$ -tuplet of real numbers.
2.
Identify how the this object is to transform under changes ofcoordinate system.
3.
Define the structure as an assignment of mathematical objectsof the type identified in item (1) to coordinate systems in such away that the objects assigned to two coordinate systems are related bythe transform identified in item (2).

Using this procedure, one can define all sorts of structures onmanifolds, of which we shall consider only two more examples here.One example is the vector field. A vector field may be defined as theassignment of a vector to every point in the manifold. Given acoordinate system, we may specify such an entity by assigning an $n$ -tuple of numbers to every point. In other words, in a coordinatesystem our vector field is represented by an $n$ -tuple of functions.Upon making a change of coordinates, these $n$ -tuples change accordingto the law presented earlier.

The second example is the metric field. Recall that in our discussionof intrinsic geometry, we considered measuring lengths and anglesalong the surface. Now, in Euclidean geometry, we may define thenotions of length and angle in terms of an inner product. A metricfield may be described as the assignment of an inner product fortangent vectors to every point of the manifold. Given a basis, aninner product can be described by a symmetric, positive-definitematrix. Hence, to define the notion of metric field, we will considerthe assignment of a symmetric, positive definite matrix of functionsto every coordinate system in such a way that the matrices assigned totwo coordinate systems are related according to the transformation lawfor inner products under a change of basis.

Once we choose a metric field on a two-dimensional manifold, itbecomes possible to study its intrinsic geometry as defined in thelast section.

0.5 Sheaves and bundles

In order to understand the toatality of all possible tangents vectors,metrics, etc. one typically collects them into larger structurescalled sheaves and bundles. To understand how such constructionsproceed, we will start by examining the fundamental example of thetangent bundle.

Last section, we desribed how a tangent vector is described in anintrinsic manner. Suppose we now want to consider the totality of alltangent vectors to a manifold. We could start by picking a point ofthe manifold and considering all tangent vectors based at that point.Since it is meaningful to make linear combinations of tangent vectorswith the same basepoint, the set of all tangent vectors with a commonbasepoint forms a vector space called the tangent space to themanifold at that point. As we saw earlier, vector spaces aremanifolds because one can impose coordinates on a vector space bychoosing a basis.

So our problem of describing the toality of all tangent vectorsreduces to the problem of describing the totality of all tangentspaces. We claim that they form a manifold. Basically, the reasonfor this is that, to specify a tangent vector, we could give thecoordinates of its basepoint with respect to a coordinate system onthe manifold and specify which vector by its coordnates with respectto a basis for the tangent space at that point. This manifold whosepoints are tangent vectors to a certain manifold is known as thetangent bundle of the orginal manifold.

0.6 Back to Erlangen

When written, this entry will show how differential geometry may beunderstood as the study of invariants of structures on manifolds underthe group of diffeomorphisms. In terms of this definition, we shallintroduce such geometries as Riemannian geometry, conformal geometry,Kahler geometry, symplectic geoemtry, contact geometry, teleparallelgeometry, gauge geometry, etc. much as Euclidean, affine, andprojective geometry were introduced above.

0.7 Differential invariants and local differential geometry

When written, this section will give the reader the flavor of the nuts and bolts of constructing the invariants which are supposed to describe geometrical objects according to the Klein’s principles. We shall mention such topics as Christoffel’s theorem and the Bianchi identitites and give some idea of the sort of topics which one considers in local differential geometry.

0.8 Global differential geometry

When written, this section will describe the Gauss-Bonnet theorem and some of its modern descendants such as characteristic classes and index theorems.

0.9 The algebraic viewpoint

When written, this entry outline how one may take the algebra of functions of a manifold as a starting point and see that differential geometric notions correspond to algebraic constructions. In fact, one may reformulate differential geometry as the study of invariants of an algebra under the action of a group of automorphisms. Using this even more general definition, one arrives at such novel topics as the differential geometry of finite-dimensional algebras and non-commutative geometry.

0.10 Infinite-Dimensional Differential Geometry

1 Differential geometry on PlanetMath

When written, this section will include links to entries on differential geometry on PlanetMath.