Smooth surface in the context of "Surface"

⭐ In the context of geometry, a surface is formally defined in several ways. Which of the following best describes how a 'smooth surface' differs from other formalized surface types?

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⭐ Core Definition: Smooth surface

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

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👉 Smooth surface in the context of Surface

A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object. It is the portion or region of the object that can first be observed and with which other objects first interact.

The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several inequivalent such formalizations that are all called surface, sometimes with a qualifier such as algebraic surface, smooth surface or fractal surface.

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Smooth surface in the context of Riemannian metric

In differential geometry, a Riemannian manifold (or Riemann space) is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds take their name from German mathematician Bernhard Riemann, who first conceptualized them in 1854.

Formally, a Riemannian metric (or just a metric) on a smooth manifold is a smoothly varying choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.

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LINKED FROM (PARENTS)
← Surface ← Riemannian metric ← Riemannian manifold
LINKS TO (CHILDREN)
Carl Friedrich Gauss → Mathematics → Differential geometry → Smooth manifold → Surface (topology) → Riemannian metric → Embedding → Euclidean space → Gaussian curvature → Isometry →