Introduction

Differential geometry is the study of shapes and spaces using the tools of calculus and linear algebra. The word differential points to differentiation: the mathematics of change, slopes, velocities, and local approximation. The word geometry points to length, angle, area, curvature, and shape. Put together, differential geometry asks a beautiful question:

If we can zoom in on a shape and measure how it changes from point to point, what can we learn about the shape as a whole?

This book begins with familiar objects: curves in the plane, curves in three-dimensional space, and surfaces such as spheres, cylinders, and tori. From there we gradually move toward a more general idea: a manifold, which is a space that may be curved or globally complicated but looks like ordinary Euclidean space when viewed very close up. This path—from curves and surfaces to manifolds—is a standard route into the subject because it lets us see the geometric ideas before we meet their most abstract form (do Carmo, 1976; Lee, 2018).

The basic idea: study a shape by looking near each point

Imagine a circle. If you stand far away, you see it as a whole closed shape. If you zoom in very closely at one point, the circle begins to look almost like a straight line. The closer you zoom, the better this straight-line approximation becomes.

This is one of the first principles of differential geometry:

A curved object can often be understood by comparing it, point by point, with a simpler linear object.

For a curve, the simpler linear object is a tangent line. A tangent line is the line that best represents the direction of the curve at a point.

For a surface, the simpler linear object is a tangent plane. A tangent plane is the plane that best represents the surface near a point.

For a manifold, the simpler linear object is a tangent space. A tangent space is the linear space of all possible first-order directions in which one can move away from a point.

The phrase first-order means “visible at the level of ordinary derivatives.” For example, if a car moves along a road, its velocity tells us its first-order behavior: where it is heading at that instant. Acceleration tells us something more refined: how the velocity itself is changing. Differential geometry repeatedly uses this pattern. First we ask for directions; then we ask how those directions change.

Curvature: the central geometric signal

A straight line does not bend. A circle bends constantly. A spiral may bend more in some places than in others. This bending is called curvature.

At a beginner level, it is helpful to think of curvature as a measurement of how quickly a geometric object changes direction. For a plane curve, curvature measures how rapidly the tangent direction turns as we move along the curve. A large circle has small curvature because it turns slowly. A small circle has larger curvature because it turns more sharply.

For example, a circle of radius \(r\) has curvature

\[ \kappa = \frac{1}{r}. \]

So a circle of radius \(10\) has curvature \(1/10\), while a circle of radius \(2\) has curvature \(1/2\). The smaller circle bends more sharply.

Surfaces have richer curvature than curves. A sphere bends the same way in every direction. A cylinder bends in one direction but not along its straight ruling direction. A saddle surface bends upward in one direction and downward in another. These differences lead to several kinds of surface curvature, especially Gaussian curvature and mean curvature, which we will study later.

One of the great discoveries of classical differential geometry is that some curvature is not merely a property of how a surface sits in surrounding space. Gauss’s Theorema Egregium showed that the Gaussian curvature of a surface can be determined from measurements made within the surface itself, such as lengths and angles, without referring to the surrounding three-dimensional space (Gauss, 1828; do Carmo, 1976). This is one reason differential geometry became a bridge from visible surfaces to abstract curved spaces.

Intrinsic and extrinsic geometry

A major theme of this book is the difference between extrinsic and intrinsic geometry.

Extrinsic geometry studies a shape as it sits inside a larger space. For example, if we study a sphere as an object in ordinary three-dimensional space, we can talk about its outward normal direction, how it bends in space, and how it appears from the outside. These are extrinsic ideas.

Intrinsic geometry studies what can be measured by someone living inside the shape. Imagine a tiny traveler living on the surface of a sphere who cannot step outside it. The traveler can measure distances along the surface, angles between paths, and areas of regions. These are intrinsic measurements.

A cylinder gives a useful first example. A sheet of paper can be rolled into a cylinder without stretching it. Distances measured on the paper do not change when it is rolled. Intrinsically, the flat sheet and the cylinder have the same local distance geometry. Extrinsically, however, they look different because the cylinder bends in three-dimensional space.

This distinction becomes especially important when we study manifolds. A manifold does not always come to us as something sitting inside a larger Euclidean space. Sometimes the space itself is the main object, and we want to study its geometry from within.

Coordinates: useful labels, not the geometry itself

To calculate, we need coordinates. A coordinate system assigns numbers to points so that we can describe positions and use algebra.

For example, a point in the plane can be described by Cartesian coordinates \((x,y)\). A point on a sphere can be described using latitude and longitude. A point on a surface may be described by two parameters \((u,v)\), where changing \(u\) and \(v\) moves us around the surface.

But coordinates are not the same thing as geometry. They are labels. The same point may have different coordinate descriptions depending on the coordinate system we choose.

For instance, in the plane, the point with Cartesian coordinates \((1,1)\) can also be described using polar coordinates \((\sqrt{2},\pi/4)\). The point has not changed. Only the description has changed.

Differential geometry teaches us to use coordinates effectively while also learning to recognize which ideas do not depend on coordinates. Length, angle, curvature, and geodesic behavior should have geometric meaning beyond a particular labeling system.

This is why the subject often moves between two styles of thought:

• Coordinate computations, where we write formulas and calculate.
• Coordinate-free thinking, where we describe objects by their geometric roles.

Both are important. Coordinates make problems computable. Coordinate-free ideas reveal what the computations mean.

From curves to surfaces to manifolds

This book follows a gradual path.

We begin with curves, because a curve is the simplest object that can bend. A parametrized curve is a curve described by a function of one variable, often interpreted as time. For example,

\[ \gamma(t) = (\cos t, \sin t) \]

describes the unit circle in the plane. As \(t\) changes, the point \(\gamma(t)\) moves around the circle. From this description we can compute velocity, speed, tangent direction, arc length, and curvature.

Next we study surfaces. A surface usually needs two parameters. For example, a sphere of radius \(1\) can be described by

\[ X(u,v) = (\sin u \cos v, \sin u \sin v, \cos u), \]

where \(u\) and \(v\) play the role of surface coordinates. On surfaces we can discuss tangent planes, surface area, normal vectors, and curvature.

Then we move to manifolds. A manifold is a space that locally resembles Euclidean space. The word locally means “near each point.” For example, the surface of the Earth is curved as a whole, but a small neighborhood around one person looks approximately flat. That local flatness is the guiding intuition behind manifolds.

A circle is a one-dimensional manifold because near each point it looks like a line. A sphere is a two-dimensional manifold because near each point it looks like a plane. More abstract spaces can also be manifolds, even when we cannot visualize them directly. Modern differential geometry studies such spaces using tangent spaces, smooth maps, differential forms, Riemannian metrics, connections, and curvature tensors (Lee, 2013; Lee, 2018).

Do not worry if these later terms are unfamiliar. They will be built slowly.

Why calculus and linear algebra are enough to begin

Differential geometry may sound advanced, but its first tools are familiar.

From calculus, we need ideas such as derivatives, partial derivatives, gradients, and the chain rule. These help us measure how functions and parametrized shapes change.

From linear algebra, we need vectors, linear maps, dot products, determinants, eigenvalues, and bilinear forms. These help us describe tangent spaces, lengths, angles, areas, and curvature in a precise way.

For example, the derivative of a curve gives a velocity vector. The dot product of two tangent vectors gives the angle between them. A matrix can describe how one tangent plane is mapped to another. Eigenvalues can identify special bending directions on a surface.

The subject becomes powerful because it combines these tools with geometric interpretation. We are not doing calculus merely to compute derivatives. We are using derivatives to understand shape.

A small example: the circle

Let us look at a familiar curve:

\[ \gamma(t) = (\cos t, \sin t). \]

This function assigns to each real number \(t\) a point on the unit circle. Its derivative is

\[ \gamma'(t) = (-\sin t, \cos t). \]

This derivative is the velocity vector. It points tangent to the circle. Its length is

\[ \|\gamma'(t)\| = \sqrt{(-\sin t)^2 + (\cos t)^2} = 1. \]

So the curve is traversed with constant speed \(1\).

Now differentiate again:

\[ \gamma''(t) = (-\cos t, -\sin t). \]

This second derivative points toward the center of the circle. It tells us that the velocity direction is constantly changing inward. That inward change is exactly the source of the circle’s curvature.

Already, with only derivatives and vectors, we have found the basic geometric structure of the circle: tangent direction, speed, acceleration, and bending.

This is the spirit of the whole book. We begin with computations, but we interpret them geometrically.

Another example: the sphere and the ant

Imagine an ant walking on a sphere. The ant cannot see the sphere from outside. It can only move along the surface and measure distances traveled.

If the ant walks straight ahead without turning left or right, its path may be a great circle, such as the equator or a longitude line. A great circle is the intersection of the sphere with a plane through the sphere’s center. Great circles are examples of geodesics on the sphere.

A geodesic is the curved-space analogue of a straight line. More carefully, a geodesic is a curve whose velocity vector is transported parallel to itself along the curve. In many familiar settings, sufficiently short pieces of geodesics are also shortest paths between nearby points, though longer pieces may stop being globally shortest (Lee, 2018).

For example, on the sphere, the shortest path from the North Pole to a point on the equator follows a longitude line. But from the North Pole to the South Pole, there are infinitely many longitude semicircles of equal length. This shows that shortest-path behavior on curved spaces can be subtler than in the plane.

What this book will train you to see

By the end of the book, you should be able to look at a geometric situation and ask productive questions:

What are the points of the space?

What does it mean to move smoothly through the space?

What are the tangent directions at a point?

How are length and angle measured?

How does curvature appear?

Which properties depend on the surrounding space, and which are intrinsic?

How do local measurements influence global shape?

These questions are more important than memorizing formulas. Formulas matter, but they are tools. The deeper goal is to develop geometric thinking: the ability to move between pictures, calculations, definitions, and meaning.

A note about abstraction

This book starts concretely because abstraction is easier when it grows from examples. We will not begin by announcing a long list of definitions about manifolds. Instead, we will first study curves and surfaces where the ideas can be seen.

When we later define a manifold, the definition will not feel arbitrary. You will already know why we need local coordinates, tangent spaces, smooth maps, and intrinsic measurements. The abstract language will organize ideas we have already met.

This is how much of mathematics develops. A concrete problem appears first. Then a pattern is noticed. Then a definition is created to capture the pattern. Then the definition becomes a tool for seeing further.

Differential geometry is rich because it follows this path again and again.

The journey ahead

The first chapters prepare the tools: linear algebra and multivariable calculus. Then we study curves, where velocity and curvature are easiest to see. After that we study surfaces, including the first and second fundamental forms, Gaussian curvature, mean curvature, geodesics, and Gauss’s Theorema Egregium.

The middle part of the book develops intrinsic surface geometry: covariant derivatives, parallel transport, and the Gauss–Bonnet theorem. These ideas show that curvature is not only local bending; it can also interact with global topology, the study of broad structural features such as holes and connectedness.

The later chapters move from surfaces to manifolds. There we meet tangent spaces, smooth maps, vector fields, differential forms, Stokes’s theorem, Riemannian metrics, connections, and curvature tensors. These ideas form a foundation for many areas of modern mathematics and mathematical physics, including Riemannian geometry and general relativity (Lee, 2018; O’Neill, 1983).

For now, the most important thing is simple:

Differential geometry studies spaces by combining local linear approximation with global geometric meaning.

If you understand that sentence, you already have the seed of the subject.

References

do Carmo, Manfredo P. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.

Gauss, Carl Friedrich. “Disquisitiones generales circa superficies curvas.” Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, vol. 6, 1828, pp. 99–146.

Lee, John M. Introduction to Smooth Manifolds. 2nd ed., Springer, 2013.

Lee, John M. Introduction to Riemannian Manifolds. 2nd ed., Springer, 2018.

O’Neill, Barrett. Semi-Riemannian Geometry: With Applications to Relativity. Academic Press, 1983.