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 ballandsocket 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
halfcrucified. 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. Mmmm! 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 GaussBonnet 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 mapmaking. Imagine that you were a sixteenthcentury
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 windswept 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
blackhole geometry may look like.

Consider now the more downtoearth 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 threedimensional 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
threedimensional FrenetSerret 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,
nonEuclidean geometry, calculus of variations,
tensor calculus, differential topology,
symplectic geometry, Finsler geometry,
deRham 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 threedimensional 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
JeanBaptiste Marie Meusnier (17541793), 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 (18161900) and Joseph A. Serret (18191885) of the
FrenetSerret formulae for describing the shape of a smooth curve in
space, and there's Pierre Bonnet (18191892) of the GaussBonnet
theorem and Joseph Bertrand (18221900) 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 (17071783), 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 oldfashioned 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 modernday 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 twodimensional 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 chockfull 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
(17921856) and János Bolyai (18021860), two names associated with
the discovery of nonEuclidean geometry. I mention them because
their ideas were important in stimulating Bernhard Riemann
(18261866) 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 ultraconservative 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 (18311916) 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 coldhearted
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 nonmathematical 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
nonEuclidean 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 "nfold extended quantity" to refer to an
ndimensional manifold.
The next section Riemann defines very verbosely in a complicated way
(remember, this is a lecture for nonmathematicians) 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 socalled 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 (18691951) 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 (19031999) 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 (19112004) 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 (18561928), Gregorio Ricci (18531925). and Tullio LeviCivita (18731941) 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
(18621943) 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 threedimensional 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 (18921977) had done something similar, but his ideas extended in a different direction.
From another angle, Albert Einstein (18701955) 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 (18781936) 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. Nondefinite 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 (18421899) who decided to carry out the ideas of
Felix Klein (18491925) 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 namedropping 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 twentiethcentury 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 GaussBonnet 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.
Suggested Reading
If you want to start having a look at what differential geometry has to offer, I propose the following bibliography:
 Differential Geometry of Curves and Surfaces. Do Carmo, Manfredo Perdigao. This book introduces differential geometry of two and threedimensional Euclidean space with relatively little prerequisites. I would call this a presentation of classical differential geometry from a modern viewpoint, since do Carmo practically gives the abstract definitions of a manifold, but by a sleight of hand specialises them to curves and surfaces. This used to be something that bothered me, but now I recognise the importance of having a firm intuitive grasp on classical differential geometry before drowning in the abstraction. Do Carmo was a student of Chern, and his exposition is clear, although it's a little clearer if you understand that he's gearing everything towards the more general study of manifolds without ever explicitly declaring so. This has become a rather standard text in the undergraduate curricula.
 A Comprehensive Introduction to Differential Geometry. Spivak, Michael. Wow, where to begin. This is a fivevolume treasure trove of diffgeo goodness. I consulted portions of the second volume for the brief historical sketch I gave above. Spivak's style is eminently readable, and he covers more ground than anyone else out there does in an introductory textbook. The prerequisites for reading these books may be a little bit higher than other books, but Spivak's other short little book, Calculus on Manifolds should be more than adequate preparation for the wonders of his comprehensive introduction.

An Introduction to Differentiable Manifolds and Riemannian Geometry. Boothby, William. I like this book because it presents modern differential geometry with all the formalism and rigour that most pleases a true mathematician. It covers all the basics of manifolds quickly and clearly, plus some more advanced topics, without ever sacrificing precision of mathematical ideas. It's a good book for the upper level undergraduate or beginning graduate student of mathematics.

Schaum's Outline of Tensor Calculus. Kay, David. Don't be fooled by the bright and colourful packaging the marketing spooks have chosen in the modern editions of Schaum's Outlines. These books are well worth your (relatively little) money, and they really are mostly all old books from the fifties, sixties and thereabouts but rebound in fancy colours. This one is especially recommended for physicists who need to get down and dirty with tensorial calculations, and for the mathematicians who want to slum with those dirty physicists.

An Introduction to Differential Geometry. Willmore, T. J. This book is probably hard to find, but it's one of my favourites. It's an old book first published in 1959 for students of British universities that does modern differential geometry the oldfashioned English gentlemanly way, if you know how I mean. It begins with subjects of classical differential geometry, but soon moves into tensor calculus and Riemannian geometry. Lots of those tensor things all around. If you want to know what Élie Cartan meant with the "debauch of indices" this is the book that best introduces the need for such debauchery and explains it surprisingly clearly. Meet the Einstein summation convention. Love the Einstein summation convention.
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://wwwgroups.dcs.stand.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.