About: Differential geometry of curves   Sponge Permalink

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In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. For example, circle in the plane can be defined as the curve γ where the vector γ(t) is always perpendicular to the tangent vector γ‘(t). Or written as an inner product

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  • Differential geometry of curves
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  • In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. For example, circle in the plane can be defined as the curve γ where the vector γ(t) is always perpendicular to the tangent vector γ‘(t). Or written as an inner product
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  • In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. For example, circle in the plane can be defined as the curve γ where the vector γ(t) is always perpendicular to the tangent vector γ‘(t). Or written as an inner product The differential properties of many classical curves have been studied thoroughly: see the list of curves for details. The main contemporary application is in physics as part of vector calculus. In general relativity for example a world line is a curve in spacetime. To simplify the presentation we only consider curves in Euclidean space, it is straightforward to generalize these notions for Riemannian and pseudo-Riemannian manifolds. For a more abstract curve definition in an arbitrary topological space see the main article on curves.
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