Riemannian geometry facts for kids

Riemannian geometry is a part of differential geometry. It studies special shapes called Riemannian manifolds. These are smooth shapes that have a special way to measure distances and angles at every point. Think of it like having a tiny ruler and protractor everywhere on the surface. This helps us measure things like angles, the length of curves, and the size of surfaces or volumes. We can then add up these small measurements to find bigger, overall measurements for the whole shape.

This type of geometry was first thought of by Bernhard Riemann in the 1800s. He wrote about it in his lecture "On the Hypotheses on which Geometry is Based." It's a big, advanced idea that builds on the study of curved surfaces in 3D space. Riemannian geometry helped bring together many ideas about curved surfaces and the paths that connect points on them (called geodesics). It also gave us ways to study shapes in higher dimensions. This field was very important for Albert Einstein's general theory of relativity. It also had a big impact on other areas of math like group theory and analysis.

What is Riemannian Geometry?

Bernhard Riemann, who first described Riemannian geometry.

Riemannian geometry was first described by Bernhard Riemann in the 19th century. It looks at many different types of geometries where measurements like distance can change from point to point. This includes types of non-Euclidean geometry, which are geometries different from the flat geometry we learn in school.

Every smooth shape can have a "Riemannian metric." This metric helps solve problems in differential topology, which is about the properties of shapes that stay the same even when the shape is stretched or bent. Riemannian geometry also helps us understand pseudo-Riemannian manifolds. These are important in general relativity, Einstein's theory about gravity and the universe.

There is a cool link between differential geometry and tiny flaws in crystals. Things like dislocations (missing lines of atoms) and disclinations (misaligned areas) in crystals can cause twists and curves, similar to how geometry works.

Here are some topics that can help you understand more:

• Metric tensor
• Riemannian manifold
• Levi-Civita connection
• Curvature
• Riemann curvature tensor
• List of differential geometry topics
• Glossary of Riemannian and metric geometry

Important Ideas in Riemannian Geometry

Here are some of the most famous ideas and theorems in Riemannian geometry. These are chosen because they are very important and elegant.

General Theorems

• Gauss–Bonnet theorem

This theorem is for 2-dimensional Riemannian shapes, like the surface of a ball. It says that if you add up the "Gauss curvature" over the whole shape, it will always equal 2π times a number called the Euler characteristic. The Euler characteristic is a number that describes the shape's topology (like how many holes it has). This theorem has a more general version for shapes in higher, even dimensions.

• Nash embedding theorems

These theorems tell us that any Riemannian manifold can be perfectly placed (or "embedded") inside a larger, flat Euclidean space. Imagine a crumpled piece of paper; this theorem says you can always smooth it out and place it perfectly flat in a bigger space without changing its internal measurements.

Geometry in Large

These theorems look at how the local properties of a shape (like its curvature) can tell us about its overall, global structure. This includes information about the shape's topology or how points behave far away from each other.

Curvature and Shape

• Sphere theorem.

If a compact (closed and bounded) Riemannian manifold has a specific type of curvature (called "sectional curvature") that is "pinched" between two values, then it is very similar to a sphere.

• Soul theorem.

If a non-compact (not closed or bounded) Riemannian manifold has non-negative curvature, it contains a special compact part called its "soul." The whole manifold looks like the "normal bundle" of this soul. If the manifold has strictly positive curvature everywhere, it is similar to flat Euclidean space.

• Cartan–Hadamard theorem

This theorem says that if a complete and simply connected (no holes) Riemannian manifold has non-positive curvature, then it is similar to flat Euclidean space. It also means that any two points in such a manifold are connected by only one shortest path (geodesic).

Ricci Curvature

• Myers theorem.

If a complete Riemannian manifold has positive "Ricci curvature," then its fundamental group (which describes the loops you can make on the shape) is finite.

• Splitting theorem.

If a complete Riemannian manifold has non-negative Ricci curvature and contains a straight line (a geodesic that is always the shortest path), then it can be thought of as a direct product of a real line and another Riemannian manifold.

• Bishop–Gromov inequality.

This inequality compares the volume of a ball in a Riemannian manifold with positive Ricci curvature to the volume of a ball in flat Euclidean space. It says the manifold's ball will have a volume at most equal to the Euclidean ball of the same size.

See also

In Spanish: Geometría de Riemann para niños

• Shape of the universe
• Basic introduction to the mathematics of curved spacetime
• Normal coordinates
• Systolic geometry
• Riemann–Cartan geometry in Einstein–Cartan theory (motivation)
• Riemann's minimal surface
• Reilly formula