# Total curvature

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In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:

## Contents

${\displaystyle \int _{a}^{b}k(s)\,ds.}$

The total curvature of a closed curve is always an integer multiple of 2π, called the index of the curve, or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.

## Comparison to surfaces

This relationship between a local geometric invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem.

## Invariance

According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.

By contrast, winding number about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.

## Generalizations

A finite generalization is that the exterior angles of a triangle, or more generally any simple polygon, add up to 360° = 2π radians, corresponding to a turning number of 1. More generally, polygonal chains that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.

The total absolute curvature of a curve is defined in almost the same way as the total curvature, but using the absolute value of the curvature instead of the signed curvature. It is 2π for convex curves in the plane, and larger for non-convex curves. [1] It can also be generalized to curves in higher dimensional spaces by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in n-dimensional space is

${\displaystyle \int _{a}^{b}\left|\gamma ''(s)\right|\operatorname {sgn} \kappa _{n-1}(s)\,ds}$

where κn1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function.

The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2π for the unknot, but by the Fáry–Milnor theorem it is at least 4π for any other knot. [2]

## Related Research Articles

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is a relationship between surfaces in differential geometry. It connects the curvature of a surface to its Euler characteristic.

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.

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.

In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The theorem was proved independently by Fáry in 1949 and Milnor in 1950. It was later shown to follow from the existence of quadrisecants.

In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : MN is an immersion if

In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928. The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by Ballmann (1990) for metric spaces of non-positive curvature and by Alexander & Bishop (1990) for general locally convex metric spaces.

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.

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

In geometry, a convex curve is a simple curve in the Euclidean plane which lies completely on one side of each and every one of its tangent lines.

In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution.

In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve.

## References

1. Chen, Bang-Yen (2000), "Riemannian submanifolds", Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, pp. 187–418, doi:10.1016/S1874-5741(00)80006-0, MR   1736854 . See in particular section 21.1, "Rotation index and total curvature of a curve", pp. 359–360.
2. Milnor, John W. (1950), "On the Total Curvature of Knots", Annals of Mathematics, Second Series, 52 (2): 248–257, doi:10.2307/1969467, JSTOR   1969467