Curvature FAQs:

Q: Is curvature any of a number of loosely related concepts in different areas of geometry?

A: Yes.

Q: Is curvature an extrinsic measure of curvature equal to half the sum of the principal curvatures, k1 + k2/2?

A: Yes, It has a dimension of length−1. Mean curvature is closely related to the first variation of surface area.

Q: Is curvature small?

A: Yes, where the curve undergoes a tight turn, the curvature is large.

Q: Is curvature thus the determinant of the shape tensor and the mean curvature is half its trace?

A: Yes.

Q: Is curvature the amount by which a geometric object such as a surface deviates from being a flat plane?

A: Yes, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context.

Q: Is curvature extrinsic and depends on the embedding?

A: Yes, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

Q: Are curvatures the eigenvalues of the shape operator?

A: Yes, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane.

Q: Is curvature available?

A: Yes, and which take the surface's unit normal vector, u into account.

Q: Is curvature normally a scalar quantity?

A: Yes, but one may also define a curvature vector that takes into account the direction of the bend in addition to its magnitude.

Q: Is curvature physical?

A: Yes.

Q: Is curvature intrinsic in the sense that it is a property defined at every point in the space?

A: Yes, and rather than a property defined with respect to a larger space that contains it.

Q: Is curvature related geometrically by the following observation?

A: Yes.

Q: Is curvature the shape operator?

A: Yes, and which is a linear operator from the tangent plane to itself.

Q: Is curvature the scalar curvature and Ricci curvature?

A: Yes.

Q: Is curvature called flat?

A: Yes.

Q: Is curvature an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface?

A: Yes, intuitively, this means that ants living on the surface could determine the Gaussian curvature.

Q: Is curvature also defined in much more general contexts?

A: Yes.

Q: Is curvature defined in analogous ways in three and higher dimensions?

A: Yes.

Q: Is curvature defined as the reciprocal of the curvature?

A: Yes.