Differential geometry is the study of smooth curvy things. Allow me to stimulate your imagination. Consider the following situations:

Consider a sheet of paper. Let us begin with a very simple idea. It is flat, but bendable, although it has a certain inflexibility. When it is flat on a desk, it has perfectly straight lines along every direction. Now pick it up, and roll up the sheet of paper, but without marking any folds. That is, you're allowed to bend the paper however you wish, but you're not allowed to fold it. Your operations have to be smooth, no edges. You should easily be able to roll into a cylinder or a cone. Observe that however you do this, at every point of your sheet of paper there will always be a direction along which perfectly straight lines exist. It seems as if you can't completely destroy the flatness of your sheet of paper if you aren't allowed to make sharp creases on it.

There's a reason for this, and there are more general things known as ruled surfaces that share this peculiar property of your humble sheet of paper.
Consider your arm. Left or right, doesn't matter; just consider an arm, any arm. In another node, ariels has described a strange situation that occurs in a sphere, but not on the sheet of paper previously considered. The ball-and-socket bone structure in our shoulders gives us a certain rotational degree of freedom in our arms, and the pair of bones in our forearms, the radius and ulna, gives our wrists the necessary rotational freedom for turning doorknobs. Consider the former degrees of freedom, but not the latter. That is, you're allowed to move the joints at your shoulder, but not rotate your wrists.

Now, try this. Hold out your arm perfectly straight, in front of you, with your hand opened, fingers together, palm down. Keeping everything rigid, rotate your arm until it is pointing straight up, as if you were asking a question in elementary school. Rotate rigidly again until your arm is again horizontal but at your side, as if you were half-crucified. Now bring your arm again in front of you again as in the beginning.

Your palm should now be pointing sideways instead of down as it originally was. You have rotated your wrist by moving your arm along a spherical triangle, but at no point did you actually use the extra rotational freedom afforded by the pair of bones in your forearm. Use it now. Keeping your arm rigid, rotate your wrist until your palm faces down. Feel the motion of muscles that you didn't use before. Because you moved your hand along a triangle lying on the sphere described by the radius of your arm, the curvature of the sphere turned your hand when you brought it back to its original position, even though you didn't rotate your wrist during these motions and kept your wrist rigid relative to the path of motion. If you had tried the same trick but moving along a zero curvature plane, your hand would have been in the same orientation when you moved it back to its original position in the plane.

This is an example of what it's like to parallel transport your hand along a spherical triangle.
Consider a shapely woman. Specifically, consider her curvatures. Part of the things that makes female curvature so stimulatingly interesting is that it is not all alike, geometrically speaking. To be sure, at breast, belly, and hip her curvatures are all quite similar, looking vaguely spherical and locally extending from her body. But consider her waist. Something interesting happens here. The curve following her waist in the vertical direction curves in a different direction than the perpendicular horizontal curve enclosing her midriff. This is different than a pair of perpendicular curves at a breast, which both curve inwards. This has the effect that at her waist, her curvature somehow bends inwards towards her body instead of away from it as it occurs further down at her hip.

What happpens, you see, is that at her hip her Gaussian curvature is positive, but at her very interesting waist it is negative. Variety is the spice of life.
Consider a cinnamon bagel with raisins. Mm-mm! Cinnamon! Let's talk more about curvature. Your bagel should have a bit of a hole in the middle of it, probably not too big, but a hole at any rate. Maybe we should have considered doughnuts instead, but that's so cliché, and I like bagels better. So, before we munch on this delicious bagel, let us examine that hole more closely. It shares a property with our shapely woman's waist, that is, curvature is negative near the hole. In fact, although our bagel is rather irregular and perhaps lumpy in some portions, it is nevertheless smooth and curvy. It has areas of positive curvature near the edges we're about to bite and areas of negative curvature near the hole.

The Gauss-Bonnet theorem tells us that the total curvature of our cinnamon bagel adds up to zero, and that this happens with any other sort of pastry (such as doughnuts) that has a hole through it. Most remarkably, a similar result holds for the total curvature of a Tim Hortons timbit (sphere), which is 4π, and the total curvature of any smooth curvy thing only depends on the number of holes the smooth curvy thing has, with each hole subtracting 4π from the total curvature.
Consider map-making. Imagine that you were a sixteenth-century cartographer entrusted with the task of giving an accurate depiction of all known Terra Firma on a flat piece of vellum. Given how your perspective of the world has recently become more broad, you are now faced with the challenge of reproducing a mostly spherical Earth on a mostly flat piece of calfskin.

You will soon run into difficulties, because just as it is impossible to flatten orange peels without tearing them or to wrap a sheet of paper around a sphere without putting creases into it, it's impossible to draw the Earth on your vellum without distorting the picture somehow, changing the apparent size of the Old and New Worlds alike. What a conundrum.

If you had been working three centuries later, you would have known that your map will be distorted because of Gauss's Theorema Egregium, that most excellent theorem, since your vellum has zero curvature but a sphere does not.
Consider the wacky ideas of a patent office clerk later in his life. Y'know, the guy with the wind-swept hair who dreamed of riding light rays. Consider what it would be like to travel across space and time to distant stars, and what it would be like to get close to a massive object such as those mysterious black holes could be.

Our patent office clerk couldn't quite figure this one out by himself, and had to ask at least one mathematician for help, but it turns out that space itself, the very medium in which we live in, is no longer so well described by the straight lines of Euclidean geometry that have served us so well in the short distances of our humble green planet. No, black holes, bend spacetime itself and give it nonzero curvature. Light always travels along paths of shortest distance, but you'll find that paths of shortest distance in the geometry of massive objects aren't going to be as straight as you might think. There will be parallel lines meeting at a point and such weirdness foreseen by Bolyai and Lobatchevsky a century earlier in a different context. How strange!

It turns out that the Riemann curvature tensor of the spacetime differential manifold describes much more of what this local black-hole geometry may look like.
Consider now the more down-to-earth experience of soap film bubbles. You might be most familiar with the situation of a free spherical bubble, but a little experimentation in a bubble bath in the spirit of childhood exploration when all the world was new is most educational, not to mention recreational and nurturing for your soul.

What happens is that Mother Nature is a relaxed lady with no interest in exerting more effort than she needs to. In this situation, it means that she absolutely refuses to make soap films experience any more surface tension than what is strictly necessary, which in turn translates into soap films taking on shapes that, at least locally, because Mother Nature doesn't always feel compelled to find the best global solution when one that work locally is good enough, minimise their surface area. You can either minimise surface area when you try to enclose a volume of air, as the soap bubbles are valiantly endeavouring, or you can minimise the surface area of soap films stretched across your hands in your bubble bath, or perhaps more practically yet boringly, stretched across narrow wires defining the boundaries of your soap film bubbles.

With a little calculus of variations, you can see that these minimal surfaces of soap films obey the remarkable requirement of zero mean curvature.
Consider, finally, the free path traced out by one of Mother Nature's creatures in three-dimensional space. Some may like to think of flying insects, avian creatures, or winged mammals, but I am a creature of water and will think of dolphins instead. This dolphin, or Darius as he prefers to be called, is equipped not only with a strong tail for propelling himself forward, but with a couple of lateral fins and one dorsal fin for controlling his direction. These give him a range of motion which he uses for exploring his native waters in the Atlantic Ocean.

Darius is a playful fellow, and sometimes he likes to see just how much he can move relying entirely on the motions of his tail and without using his fins. He restricts his motion to the vertical strokes of his tail and the accompanying undulations this necessitates in the rest of his body. It turns out that this still gives him quite a broad range of motion, except that the paths he can trace out in this manner, winding as they may be, are restricted to lie within a vertical plane. When he has had enough of this sport, Darius tilts his body his body until his belly now faces sideways, and he swims in a different direction, outside of the plane in which he had originally confined himself for his amusement.

What Darius has discovered in his sinuous exploration is that if he keeps his torsion zero by not tilting his body with his fins, then the curve traced out by his motion is confined to a plane, just as the three-dimensional Frenet-Serret formulae predicted that it would be.

Needless to say, the above considerations are all situations proper to differential geometry.

Affectionately Known as Diffgeo

Differential geometry is the branch of geometry that concerns itself with smooth curvy objects and the constructions built on them. Differential geometry studies local properties such as measuring distance and curvature in smooth objects, or global properties such as orientability and topological properties.

But there is so much more to say about it than that. The term "differential geometry" often designates a broad classification of diverse subjects that are difficult to categorise separately, because interaction between these subjects is often too strong to warrant a separate study. Other terms associated with differential geometry, some used as synonyms for "differential geometry", some considered to be subdivisions of the subject, and others simply closely related are surface theory, theory of curvature, differential manifolds, Riemannian manifolds, global geometry, non-Euclidean geometry, calculus of variations, tensor calculus, differential topology, symplectic geometry, Finsler geometry, de-Rham cohomology, and general theory of relativity, to mention a few.

A first approximation to understanding what differential geometry is about is understanding what it is not about. Differential geometry contrasts with Euclid's geometry. The latter most often deals with objects that are straight and uncurved, such as lines, planes, and triangles, or at most curved in a very simple fashion, such as circles. Differential geometry prefers to consider Euclidean geometry as a very special kind of geometry of zero curvature. Nonzero curvature is where the interesting things happen.

A historical perspective may clarify matters. Differential geometry has its roots in the invention of differential and integral calculus, and some may say that it started even before that. If you've done mathematics in a lycée, gymnasium, vocational school, or high school, you arguably have already seen some rudiments of differential geometry, but probably not enough to give you a flavour of the subject. The study of conic sections, parabolas, ellipses, and hyperbolas spurs the imagination to ask questions proper to differential geometry. The real fun begins when we introduce the derivative or differential and start wondering about what the various derivatives or differentials of certain objects tell us about these objects.

Early Trailblazers

Historically, it might be possible to divide differential geometry into classical and modern, with the line of demarcation drawn somewhere across Bernhard Riemann's inaugural lecture given in Göttingen. Classical differential geometry begins with the study of curved surfaces in space, such as spheres, cones, cylinders, hyperbolic paraboloids, or ellipsoids. A key notion always present in differential geometry is that of curvature. A desire to define a notion of curvature of surfaces leads us to a simpler problem: the curvature of curves. The real defining characteristic of classical differential geometry is that it deals with curves and surfaces as subsets contained in Euclidean space, and almost invariably only considers two and three-dimensional objects.

Early classical differential geometry is characterised by a spirit of free exploration of the concepts that the invention of calculus now provided mathematicians of the day. The intuition of infinitesimals was used without any restraint for what its real meaning could be. Curves and surfaces were explored without ever giving a precise definition of what they really are (precise in the modern sense). For a modern reader, reading the classical texts therefore presents quite a challenge.

There are lots of mathematicians whose names are associated with classical differential geometry. There is Olinde Rodrigues (1794 - 1851?), a figure that history has clad in mystery but whose name survives in a theorem that gives necessary and sufficient conditions for a line on a surface to be a line of curvature. There is Jean-Baptiste Marie Meusnier (1754-1793), also a relatively obscure figure in the history of mathematics were it not for his theorem about normal curvatures of a surface. A bit later on, there's Jean F. Frenet (1816-1900) and Joseph A. Serret (1819-1885) of the Frenet-Serret formulae for describing the shape of a smooth curve in space, and there's Pierre Bonnet (1819-1892) of the Gauss-Bonnet theorem and Joseph Bertrand (1822-1900) of the Bertrand curves. The French school tradition of differential geometry extended well into the twentieth century with the emergence of an eminence such as Élie Cartan. And there's Euler (1707-1783), who is associated with every branch of mathematics that existed in the eighteenth century.

Euler can probably be creditted for much of the early explorations in differential geometry, but his influence isn't quite as profound as the reverbarations that Karl Friedrich Gauss's (1777 - 1855) seminal paper Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) (1827) propagated through the subject. Gauss's paper written in Latin, a practice that was already old-fashioned in the nineteenth century, gives us an almost modern definition of a curved surface, as well as a definition and precise procedures for computing the curvature of a surface that now bears his name. He also defines the first and second fundamental forms of a surface, and the importance of the first has survived to modern-day differential geometry in the form of a Riemannian metric in Riemannian geometry. Using these concepts, and the intrinsic property of the first fundamental form, which only depends on the surface itself, but not in how this surface is placed in the surrounding Euclidean space, he proves the theorema egregium, that remarkable theorem over which, as a beloved professor of mine once colourfully described it, "Gauss lost his pants when he saw this." The theorema egregium points out the intrinsic property of the Gaussian curvature, since it is invariant by isometries such as the folding of our sheet of paper back up there in the examples.

We have retained much of Gauss's notation to this day, such as using E, F, and G for denoting the coefficients of the first fundamental form when dealing with two-dimensional surfaces immersed in three dimensional space. Perhaps it is also in the spirit of this paper that when doing classical differential, we submerge ourselves in lengthy calculations. Well, scratch that, because modern differential geometry is still chock-full of calculations, especially when doing tensor calculus, and then we have what Élie Cartan has called "the debauch of indices". It's just that calculations in classical differential seem more necessary because nobody had stepped back from the sea of details yet and tried to understand the underlying abstraction.

I should mention two more important figures in the development of classical differential geometry, although their work was, strictly speaking, not differential geometry at the time, although it can be subsumed under the umbrella of differential geometry with the modern viewpoint. I am speaking of Nikolai Ivanovich Lobachevsky (1792-1856) and János Bolyai (1802-1860), two names associated with the discovery of non-Euclidean geometry. I mention them because their ideas were important in stimulating Bernhard Riemann (1826-1866) to the abstract definition of a differential manifold, where all modern differential geometry takes place.

An inaugural address promises bold new directions of exploration.

On June 10, 1854, Bernhard Riemann treated the faculty of Göttingen University to a lecture entitled Über die Hypothesen, welche der Geomtrie zu Grunde liegen (On the Hypotheses which lie at the foundations of geometry). This lecture was not published until 1866, but much before that its ideas were already turning (differential) geometry into a new direction.

The story of how that lecture was conceived is an interesting one, and I shall summarise it as it appears in Michael Spivak's second volume of his A Comprehensive Introduction to Differential Geometry. Riemann was seeking the position of Privatdocent, a lecturer without a fixed salary whose income is determined by the number of students that attend his lectures. For this purpose, he had to propose three topics from which his examiners would choose one for him to lecture on. The first two were on complex analysis and trigonometric series expansions, on which he had previously worked at great length; the third was on the foundations of geometry. He had every reason to suspect that his examiners would choose one of the first two, but Gauss decided to break tradition (a rare decision for the ultra-conservative Gauss) and instead chose the third, a topic that had interested him for years. At the time, Riemann was investigating the connection between electricity, light, magnetism, and gravitation, in addition to being an assistant at a mathematical physics seminar, and the strain of having to deliver a lecture on a subject he hadn't fully prepared strained him enough to give him a temporary breakdown. He recovered, and delivered his lecture.

Dedekind (1831-1916) later records how upon hearing Riemann's inaugural address, Gauss sat through the lecture "which surpassed all his expectations, in the greatest astonishment, and on the way back from the faculty meeting he spoke with Wilhelm Weber, with the greatest appreciation, and with an excitement rare for him, about the depth of the ideas presented by Riemann." Riemann was, of course, admitted.

So what was the lecture about? What could possibly move cold-hearted Gauss to such enthusiasm? There are three major important bits. For a modern reader, Riemann's address is hard to read, especially because he tried to write it for a non-mathematical audience! (A word of caution about trying to dumb down what isn't dumb: generally a bad idea, since neither the dumb nor the smart will understand.) In the preface, he gives a plan of investigation, where he seeks to better understand the properties of space in order to understand the non-Euclidean geometries of Bolyai and Lobachevsky. In modern parlance, what he attempts to do here is to exhort his listeners to separate the topological properties (shape without distance) of space from the metric properties (distance measurements). He says that if we can give space different metric properties, than different versions of the parallel postulate can arise with the same basic underlying topology of space.

In the first section beyond the preface, Riemann is trying to define the concept of a manifold, which generally speaking is this abstraction of space without distance, but that still looks like Euclidean space when you take out your microscope and peer very closely at it. He sees no particular reason to restrict manifolds to have only three dimensions, and Spivak's translation of Riemann often writes "n-fold extended quantity" to refer to an n-dimensional manifold.

The next section Riemann defines very verbosely in a complicated way (remember, this is a lecture for non-mathematicians) what a reasonable way to measure length on a manifold can be, but with enough freedom to assign different ways of length measurement that vary locally. He accomplishes this by measuring the lengths of curves by integrating the tangent vectors of these curves and scaling this integration by a function that can change smoothly over each point in the manifold. This is precisely the modern notion of a Riemannian metric, and manifolds equipped with such a metric are known as Riemannian manifolds. He goes on to give some mathematical results of what properties this metric must satisfy, and he restricts himself to a special kind of metric (dropping some of his restrictions lead Finsler in 1918 to the study of so-called Finsler metrics and to modern Finsler geometry, a fertile area of modern research).

In the third and final section of this brief but dense lecture, Riemann ponders what possible applications his ideas could have for modelling the space we live in, that is, applications to physics. It would be too much to conjecture that Riemann in any way anticipated the way that this geometry would be used in the twentieth century by Albert Einstein during his development of the general theory of relativity, but Riemann did believe that certain physical experiments could be carried out in order to better ascertain what the geometry of space should be like. This is not entirely a novel idea, dropping the assumption that Euclidean geometry is the perfect geometry for describing our universe, since Gauss earlier had already attempted to determine the possible geometry of space by measuring the angles of a triangle formed by three mountaintops, although his results led him to conclude that at least within experimental error, our geometry is Euclidean and the angles of a triangle add up to 180 degrees.

It took differential geometers close to fifty more years to fully develop Riemann's ideas and cement the notions of a manifold and a Riemannian metric. In a sense, research for describing the geometry of spacetime is still underway by astrophysicists, and Riemann's ambitions in the third section of his inaugural address are not yet completely realised. It is undeniable that Riemann brought differential geometry a modern firm footing on differential manifolds and that his ideas guided research perhaps until this very day.

The twentieth century: A cornucopia of ideas and the physicists take notice.

During the twentieth century, areas of study in differential geometry expanded at an explosive rate. During the late nineteenth century, the physicists had developed the theory of electromagnetism to a clear refinement with vector calculus that mathematicians such as the French Élie Cartan (1869-1951) later polished into the abstraction of differential forms and integration on manifolds. Classical integral theorems were subsumed under one roof of generalisation such as the modern and general version of Stokes' Theorem. These differential forms lead others such as Georges de Rham (1903-1999) to link them to the topology of the manifold on which they are defined and gave us the theory of de Rham cohomology. Later on, influential differential geometers such as the worldly Chinese mathematician S. S. Chern (1911-2004) a student of Cartan, refined and spread the ideas of differential geometry across the globe (and is probably largely responsible for the proliferation of differential geometry in Brazil, Argentina, and other parts of Latin America).

The Italians Luigi Bianchi (1856-1928), Gregorio Ricci (1853-1925). and Tullio Levi-Civita (1873-1941) clarified the notions of differentiation on a manifold and how to move from one tangent space to another in a sensible way via their development of the tensor calculus. The German David Hilbert (1862-1943) has a stab at some theorems of global differential geometry, and proves that a surface of constant negative curvature on which we can model hyperbolic geometry, such as the pseudosphere, cannot fit completely in three-dimensional space without singularities. The American John Milnor (1931- ) realises that differential geometry has something to offer to topology and gives birth to the subject of differential topology. Earlier another American, Marston Morse (1892-1977) had done something similar, but his ideas extended in a different direction.

From another angle, Albert Einstein (1870-1955) started to see that he needed a new theory of geometry if he was to generalise his theory of relativity to the case of noninertial frames of reference. He recruited the help of mathematician friend and former classmate Marcel Grossmann (1878-1936) who found the necessary tools in the tensor calculus that the Italian school of differential geometry had created earlier. Once physics found applications for the differential geometry that mathematicians had been developing for so long, it started to contribute to the subject and develop its own tradition and schools.

The intervention of the physicists enriched and complicated the subject immensely, with mathematicians sometimes working in parallel with the physicists' traditions, sometimes intersecting, sometimes not, as if trying themselves to imitate the same variations of the parallel postulate that their study of manifolds now afforded them. Non-definite metrics such as the Minkowski metric that describes the geometry of spacetime gained prominence. From a different direction, classical and analytical mechanics and its study of mechanical system lead to the birth of symplectic geometry. Physics has given a wealth of ideas to differential geometry.

Yet another tributary to this river of dreams came a little earlier in the late 19th century from the Norweigian Sophus Lie (1842-1899) who decided to carry out the ideas of Felix Klein (1849-1925) and his Erlanger Programm and consider continuous, differentiable even, groups that could tell us something about the symmetries of the manifolds under scrutiny, these groups also manifolds in their own right themselves. His Lie groups are an important area of modern research in themselves.

There are many, many, many more mathematicians and physicists that contributed to modern differential geometry throughout the twentieth century, and it is impossible to mention them all. Here I have merely attempted to mention some of the most famous figures and their most outstanding contributions. It is even difficult to categorise all of differential geometry, as the subject has grown into many diverse fields, that sometimes it is even difficult to say whether they are related fields or completely different altogether.

Right. Sorry for all the name-dropping and jargon above. I want to point out that there is still one common thread underlying all of these various currents of thought, though. Differential geometry is the study of smooth curvy things. Remember that. Even if there are many different ways to look at the same curvy thing, it's still a curvy thing in the end.

Diffgeo for the modern student of mathematics

If you want to get initiated into the study of differential geometry today, you would do best to first have a good grasp of linear algebra and vector calculus. Knowledge of some modern analysis, enough to understand the fundamentals of metric and topological spaces, will also be quite handy, though sometimes not essential. With such preparation, you should be ready to take an undergraduate course in differential geometry. Typically, a first course presents classical differential geometry in two and three dimensions using various modern lenses in order to better see the development of ideas, and it might dip its toes into more modern subjects such as the abstract definition of a differential manifold.

These things are of course highly variable, but early on in your studies of differential geometry, you should also see something about integration of differential forms (a twentieth-century topic when done with the proper modern abstraction), differentiation on manifolds, a hint at the connections between the topological properties of a manifold and its curvature (such as the Gauss-Bonnet theorem). You might also see some of the geometrical constructions that can be done on a manifold, such as (tangent) bundles. It's also possible that you'll have to learn some tensor calculus in order to formalise computations on manifolds, especially if you're approaching the subject from a physicist angle, although nothing is set in stone, and mathematicians may be required to know how to deal conveniently with tensors and tensor fields just the same.

Differential geometry is an attractive object of study. It appeals to our geometric intuition, which some have argued is the true source of all of mathematics, and it's overflowing with beautiful theorems and surprising results. There are lots of abstractions to complement our intuition, and with a little bit of effort they can all be juxtaposed to rather tangible objects that can be used to verify their validity and purpose. It even has applications for people as practical as engineers in control theory, since the configuration space of a mechanical system can be succinctly described as a manifold of dimension equal to the degrees of freedom of the system, and in computer graphics.

It's quite simply gorgeous. Definitely one of my favourite branches of mathematics.

References

In addition to the books mentioned above which I briefly consulted for writing this node, I also consulted The MacTutor History of Mathematics archive (http://www-groups.dcs.st-and.ac.uk/~history/) which has become a standard online reference for biographies of mathematicians, plus the courses, lectures, workshops, and conferences I have attended in differential geometry, and although I wish I could call myself a differential geometer, I have to admit that I'm still a newbie in the subject.

manifold	demonstration of curvature	Blind faith in Science is just as bad as blind faith in Religion	The Teenage Stereotype of Technology
String Theory	The Parable of the Coffee Cup and the Donut	Geodesic	debauch of indices
minimal surface	abstract algebra	quantum field theory	theorema egregium
Riemann Curvature Tensor	topological vector space	metric tensor	mathematics
Tullio Levi-Civita	Sophus Lie	spinor	Series 7 Exam
The Feynman Lectures on Physics	horocycle	Mouth breathing	sheaf theory