**What is Riemannian Geometry?** - **Beyond Euclid:** Riemannian geometry is a branch of differential geometry that generalizes the classic Euclidean geometry of flat spaces to curved surfaces and higher dimensions. It was pioneered by [[Bernhard Riemann]] in the 19th century. - **The Main Idea:** Riemannian geometry studies spaces equipped with a Riemannian metric. This metric is a smoothly varying way to define distances and angles at each point on a surface (or in higher dimensions, a manifold). - **Think Curvature:** Imagine measuring distances on the Earth's surface (curved) vs. on a flat table. The way we calculate distances and directions on the Earth is different due to its curvature. Riemannian geometry provides the mathematical framework to describe such curved spaces. **Key Concepts** - **Manifolds:** The fundamental objects of study in Riemannian geometry. A manifold is a space that locally looks like Euclidean space (flat) but may have a more complex global shape (like a sphere or donut). - **Riemannian Metric:** This defines inner products (a way to measure angles and lengths) on the tangent space at each point of a manifold. It changes smoothly from point to point, allowing for the measurement of curved distances. - **Geodesics:** The generalization of "straight lines" on curved spaces. Geodesics are the shortest paths between two points on a Riemannian manifold. - **Curvature:** Measures how much a space deviates from flatness. Riemannian geometry provides different ways to quantify curvature, including Gaussian curvature and sectional curvature. **Significance** - **General Relativity:** Riemannian geometry is the mathematical foundation of Einstein's General Theory of Relativity, where gravity is understood as the curvature of spacetime caused by mass and energy. - **Beyond Physics:** Applications in optimization problems, machine learning, computer graphics, and other fields where working with complex, potentially curved data is important. **Example** Imagine a sphere. While a sphere is embedded in 3D-space, its surface is intrinsically a 2D curved space. Riemannian geometry allows us to: - Measure distances between two points on the sphere along the surface itself (not cutting through 3D space). - Determine the shortest routes between points on the sphere (which are arcs, not straight lines). - Understand how the sphere's curvature affects geometric properties. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Riemann geometry") sort modified desc, authors, title ```