The Gaussian curvature is the product of the two principal curvatures . The Ricci curvature is essentially an average of curvatures in the planes including . Spinal curvatures of kyphoscoliosis or lumbar lordosis may be present. This rolling can be done at different angles and curvatures resulting in different nanotube properties. The primary curves (thoracic and sacral curvatures) form during fetal development. Severe curvatures that rapidly progress may require surgery with spinal rod placement and spinal fusion. We see that projective transformations don't mix Gaussian curvatures of different sign. The eigenvalues are the principal curvatures of the surface. The eigenvalues of H are proportional to the principal curvatures of D. Long-term consequences include permanent curvatures or disfiguration of the long bones, and a curved back. Capos come in different sizes and shapes for different instruments and fretboard curvatures. Skeletal problems may also be present, including abnormal curvatures of the spine. The sphere and the plane have different Gaussian curvatures, so this is impossible. The eigenvalues of are just the principal curvatures and at . Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.