**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 lengthâˆ’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.