La Fantastica Storia della GEOMETRIA DIFFERENZIALE Parte 2: La Curvatura di GAUSS

In this second episode dedicated to the history of differential geometry, we continue the journey begun with non-Euclidean geometries and Harriot's formula for triangles on a sphere. Starting with the angular excess of spherical triangles, we see how Gauss took the decisive step: transforming a global property of the sphere into a local definition of curvature valid for any surface. This gave rise to Gaussian curvature, defined as the limit of the angular excess per unit area when a geodesic triangle narrows around a point. We then analyze the geometric meaning of positive and negative curvature, the role of surfaces of constant curvature, and the ideas that led to the famous Gauss–Bonnet local theorem, where the angular excess of a triangle is interpreted as the total curvature contained within it. In the final section, we delve deeper into the mathematics of the problem, studying the geodesic circle on the sphere, its circumference, the spherical cap, and how deviation from Euclidean formulas allows us to directly measure the curvature of space. A journey through history, geometric intuition, and the first calculations that led to the birth of modern differential geometry. #geometry #differentialgeometry #gauss #gaussbonnet #curvature #euclid #mathematics #math #mathematics #topology #riemann #lobachevsky #bolyai #surfaces #science #popularization #physics #curvedspace #historyofmathematics References: Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts ⭐️Subscribe and board the science spaceship ⭐️:    / @yousciences   Where to find me 💎: ➤YOUSCIENCES: https://www.yousciences.it/ ➤OFFICIAL WEBSITE: https://www.giuseppesottile.it/ ➤INSTAGRAM: @yousciences @___giux___ ➤FACEBOOK:   / giuseppe.sottile.56   ➤LINKEDIN:   / giuseppe-sottile-3a8599b0   ➤TELEGRAM: https://t.me/joinchat/abNZYJqE7MVlODA0 ➤TIKTOK: https://www.tiktok.com/@yousciences?l... 📽️ VIDEO STREAM ALL: https://www.yousciences.it/videostream/ https://www.giuseppesottile.it/video.php Chapter Summary: 00:00 Overture 02:00 Gaussian Curvature 03:00 Surfaces with Constant Positive Curvature 04:12 Pseudosphere 05:12 The Gauss-Bonnet Theorem 07:50 Spherical Cap Example Executive Production by GIUX