Differential geometry

We can study smooth shapes.

Some shapes are flat like a table. Other shapes curve like a ball or a hill.

We can measure how much a shape bends. This is called curvature.

People use math to find the shortest path on a curve.

Long ago, thinkers studied the shape of the Earth. They learned how to make maps. This helped them understand our world.

Math like this helps us study space. It even helps us learn about stars and tiny particles. It is used in many ways today.

Shapes are all around us!

Math can study smooth shapes. Some shapes are flat like a table. Other shapes curve like a ball. This field of math is called differential geometry.

It looks at how shapes bend and curve. It also looks at smooth spaces called manifolds.

People have studied these shapes for a long time. Ancient Greek thinkers studied the shape of the Earth. They wanted to find the shortest path between two points. On a round Earth, this path is a great circle.

Later, math tools called calculus helped people measure curves. They could find the exact way a line bends. This bend is called curvature.

In the 1800s, math changed a lot. Carl Friedrich Gauss studied curved surfaces. He found ways to measure shapes from the inside. Other thinkers found new types of math. This is called non-Euclidean geometry. It shows that math can work in many ways.

Today, this math is very useful. Albert Einstein used it to explain how space works. It helps scientists study stars and tiny particles. It also helps with computer graphics and making maps.

Differential geometry is a special branch of math. It studies smooth shapes and smooth spaces. These spaces are often called manifolds. You can think of a manifold as a surface that is smooth rather than bumpy.

This math looks at how we measure things like size and distance on these shapes. It can also look at angles or the volume inside a shape. Different types of geometry focus on different things. For example, one kind focuses only on angles. Another kind focuses on how to calculate volume. This math helps us understand the very rules of space.

To study these shapes, mathematicians use many different tools. They use things like vector calculus and linear algebra. These tools help them describe how a curve bends or turns.

One important idea is curvature. Curvature tells us how much a shape curves at a specific point. You can see this by looking at an osculating circle. This is a circle that fits perfectly against a curve. Another idea is the geodesic. A geodesic is the shortest path between two points on a surface. On a flat table, a geodesic is a straight line. On a round ball, the shortest path is a great circle.

People have been curious about shapes for a very long time. Ancient Greek mathematicians studied the shape of the Earth. Around 200 BC, Eratosthenes calculated the circumference of the Earth. Around 150 AD, Ptolemy used a method called stereographic projection to make maps. Later, in the 1600s, new math tools were born. Isaac Newton and Gottfried Leibniz developed calculus. This allowed people to study curves much more precisely. Pierre de Fermat and the Bernoulli brothers also used these new ideas. They began to measure how lines bend and where they change direction.

Many famous thinkers added to this field over the centuries. Leonhard Euler studied how objects move along surfaces. He found the first equations for geodesics in the 1700s. In the 1800s, Carl Friedrich Gauss did some of the most important work. He wrote a huge book about curved surfaces in 1827. Gauss discovered how to measure a shape from the inside. This is called intrinsic geometry. Around the same time, Nikolai Lobachevsky and János Bolyai found hyperbolic geometry. This showed that math could work in ways that were different from flat geometry.

Today, differential geometry is used in many different ways. Albert Einstein used this math for his theory of general relativity. This theory explains how gravity works in space. Physicists also use it to study tiny particles in quantum field theory. It is not just for space and stars, though. Engineers and computer scientists use it too. It helps with computer graphics and computer vision. It is even used in chemistry and economics. It even helps with modern machine learning.

Differential geometry is a mathematical discipline focused on the study of smooth shapes and spaces. These smooth spaces are formally known as manifolds. A manifold is a space that is smooth rather than jagged or bumpy. Mathematicians use specialized tools to analyze these structures, including vector calculus, linear algebra, and multilinear algebra. The field seeks to define geometric structures on these manifolds. A geometric structure provides a way to measure properties like size, distance, shape, or volume.

Different branches of geometry focus on different specific measurements. In Riemannian geometry, the focus is on specifying distances and angles. Symplectic geometry is used when one needs to compute volumes. Conformal geometry is a branch where only angles are specified. In gauge theory, certain fields are assigned to the space. Differential geometry is also closely linked to differential topology. Differential topology studies the properties of differentiable manifolds that do not depend on additional geometric structures. It is also related to geometric analysis, which explores the geometric aspects of differential equations.

To understand how these shapes behave, mathematicians study concepts like curvature and geodesics. Curvature describes how much a curve or surface bends at a specific point. One way to visualize this is through an osculating circle. An osculating circle is a circle that fits perfectly against a plane curve.

Another vital concept is the geodesic. A geodesic is the shortest path between two points on a surface. For example, a straight line is a geodesic on a flat plane, but a great circle is the geodesic on a sphere.

The history of this field reaches back to classical antiquity. Ancient Greek mathematicians studied spherical geometry to understand the Earth. Around 200 BC, Eratosthenes calculated the circumference of the Earth. Around 150 AD, Ptolemy introduced stereographic projection to help map the Earth's shape. Even in Euclid's time, it was understood that a straight line is the shortest distance between two points. This principle led to the realization that great circles are the shortest paths on the Earth's surface. In the 1500s, Gerardus Mercator developed a map projection. Mercator understood that his map was conformal, meaning it preserved angles. However, he noted that the shortest paths on Earth appeared as curved lines on his flat map.

Modern differential geometry began to take shape after the invention of calculus in the 1600s. Isaac Newton and Gottfried Leibniz developed calculus, which provided the tools for rigorous study. Pierre de Fermat, Newton, and Leibniz investigated plane curves to find points of inflection. The Bernoulli brothers, Jacob and Johann, also made early contributions using infinitesimals. They helped calculate tangents to curves and identified the radius of osculating circles. This provided the first analytical formulas for curvature. Later, Alexis Clairaut studied space curves at age 16. He introduced the idea of tangent spaces and the concept of principal curvatures.

In the 1700s, Leonhard Euler made massive contributions to the field. He derived the first analytical geodesic equation. Euler also introduced intrinsic coordinate systems, which started the theory of intrinsic geometry. He realized that a mass moving on a surface without external force would follow a geodesic. This idea was an early precursor to general relativity. In 1760, Euler proved a theorem relating the curvature of a space curve to principal curvatures. Other mathematicians, like Gaspard Monge and Charles Dupin, further developed the theory of surfaces and curvature.

The 1800s marked a turning point when the field became a distinct area of study. Carl Friedrich Gauss published his major work on curved surfaces in 1827. Gauss is often called the inventor of non-Euclidean geometry and intrinsic differential geometry. He introduced the Gauss map, Gaussian curvature, and the fundamental forms. He also proved the Theorema Egregium, which showed that Gaussian curvature is intrinsic. During this same era, Nikolai Lobachevsky and János Bolyai independently discovered hyperbolic geometry. This proved that consistent geometries could exist outside the standard Euclidean model.

Today, differential geometry is essential to many scientific fields. Albert Einstein used its language to develop the theory of general relativity. Physicists also use these concepts to build quantum field theory and the Standard Model. Beyond physics, the field applies to chemistry, economics, and engineering. It is used in control theory and computer graphics. Computer vision also relies on these geometric principles. Most recently, differential geometry has found new applications in the field of machine learning.