## Science 2

Ut aliquet tempor quam, sit amet congue libero dignissim viverra. Proin rhoncus enim ac tincidunt tincidunt. Duis sapien magna, facilisis vitae dapibus at, lacinia quis turpis. Morbi orci tortor, tincidunt a tempor ullamcorper, volutpat sed ante. Etiam eu erat at arcu euismod luctus. Ut gravida ipsum augue, a tempus metus porta quis. Nam ultricies sit amet nisl sed aliquam. Nam eros lectus, facilisis sed magna quis, iaculis tempus massa. Aenean quis malesuada metus. Mauris venenatis pellentesque ligula ut euismod.

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen”^{[1]} (“On the Hypotheses on which Geometry is Based”). It is a very broad and abstract generalization of the differential geometry of surfaces in **R**^{3}. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein‘s general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

#### Associated publications

- PDF download:

- PDF download:

- PDF download: