Title Normal Curvature, Geodesic Curvature and Gauss Formulas MATH 2040 December 20, 2016 MATH 2040 Normal and Geodesic Curvature December 20, 2016 1 / 12
Readings Readings Readings: 4.4 MATH 2040 Normal and Geodesic Curvature December 20, 2016 2 / 12
The Normal Space Definition Normal and Geodesic Curvature Let M be a surface with normal vector n(p) at p M. The normal space to M at p is N p M = {λn(p) : λ R}. Note: it should be pictured as a line perpendicular to M at through p. The origin is p. MATH 2040 Normal and Geodesic Curvature December 20, 2016 3 / 12
The Normal Space Definition Normal and Geodesic Curvature Let M be a surface with normal vector n(p) at p M. The normal space to M at p is N p M = {λn(p) : λ R}. Note: it should be pictured as a line perpendicular to M at through p. The origin is p. The set of vectors at p decomposes into a pair of in T p M and N p M: R 3 = T p M N p M. In other words, any vector v based at p can be written v = v tan + v norm where v tan T p M and v norm N p M. MATH 2040 Normal and Geodesic Curvature December 20, 2016 3 / 12
Normal and Geodesic Curvature Decomposition of acceleration of a curve Let γ(s) be a unit speed curve. Recall that n is normal to the surface. The intrinsic normal of γ is S = n T. MATH 2040 Normal and Geodesic Curvature December 20, 2016 4 / 12
Normal and Geodesic Curvature Decomposition of acceleration of a curve Let γ(s) be a unit speed curve. Recall that n is normal to the surface. The intrinsic normal of γ is S = n T. The acceleration vector γ (s) decomposes into normal and tangent components: γ (s) = X(s) + V(s) X(s) T γ(s) M, V(s) N γ(s) M. Fact: X(s) is parallel to S(γ(s)). MATH 2040 Normal and Geodesic Curvature December 20, 2016 4 / 12
Normal and Geodesic Curvature Normal and geodesic curvatures Definition Let γ(s) be a unit speed curve in a surface M. The normal curvature of γ at s is κ n (s) = γ (s), n(γ(s)) and the geodesic curvature of γ at s is κ g (s) = γ (s), S(γ(s)). MATH 2040 Normal and Geodesic Curvature December 20, 2016 5 / 12
Normal and Geodesic Curvature Normal and geodesic curvatures Definition Let γ(s) be a unit speed curve in a surface M. The normal curvature of γ at s is κ n (s) = γ (s), n(γ(s)) and the geodesic curvature of γ at s is κ g (s) = γ (s), S(γ(s)). In this case we can always write γ (s) = κ(s)n(s) = κ n (s)n(γ(s)) + κ g (s)s(γ(s)). Notation: we sometimes shorten n(γ(s)) to n(s), etc. MATH 2040 Normal and Geodesic Curvature December 20, 2016 5 / 12
Gauss s Equations Second fundamental form part I Notation: Denote 2 x u i u j (u1, u 2 ) = x ij (u 1, u 2 ). MATH 2040 Normal and Geodesic Curvature December 20, 2016 6 / 12
Gauss s Equations Second fundamental form part I Notation: Denote 2 x u i u j (u1, u 2 ) = x ij (u 1, u 2 ). Definition Let x be a coordinate patch of a surface M. The coefficients of the second fundamental form are Note: L ik = L ki. L ik = x ik, n. MATH 2040 Normal and Geodesic Curvature December 20, 2016 6 / 12
Gauss s Equations Second fundamental form part II Definition Let x be a coordinate patch of a surface M. The second fundamental form is the map on the tangent space II p : T p M T p M R given by II p (X, Y) = i,k L ik X i Y k. for X = X 1 x 1 + X 2 x 2, Y = Y 1 x 1 + Y 2 x 2. Note: It is symmetric; that is II p (X, Y) = II p (Y, X). MATH 2040 Normal and Geodesic Curvature December 20, 2016 7 / 12
Christoffel symbols Gauss s Equations Definition Let x be a coordinate patch on a surface M. The Christoffel symbols of M with respect to x are Γ k ij = m xij, x m g mk. MATH 2040 Normal and Geodesic Curvature December 20, 2016 8 / 12
Christoffel symbols Gauss s Equations Definition Let x be a coordinate patch on a surface M. The Christoffel symbols of M with respect to x are Γ k ij = m xij, x m g mk. Another version of this formula is xij, x m = Γ k ij g km. k MATH 2040 Normal and Geodesic Curvature December 20, 2016 8 / 12
Gauss s Equations Gauss s formulas Theorem (Gauss s formulas) Let x be a coordinate patch on a surface M. Then x ij = L ij n + k Γ k ij x k. Note that this is a decomposition of x ij onto tangential and normal components. MATH 2040 Normal and Geodesic Curvature December 20, 2016 9 / 12
Gauss s Equations Geodesic and normal curvature in terms of Christoffel symbols Theorem Let x be a coordinate patch on a surface M, and let γ(s) = x(γ 1 (s), γ 2 (s)) be a unit speed curve on M. Then κ n = i,j L ij L ij dγ i ds dγ j ds and κ g S = k d 2 γ k ds 2 + i,j Γ k ij dγ i ds dγ j x k. ds MATH 2040 Normal and Geodesic Curvature December 20, 2016 10 / 12
Intrinsic quantities Intrinsic quantities We say that a quantity is intrinsic to a surface if it can be written entirely in terms of the first fundamental form. MATH 2040 Normal and Geodesic Curvature December 20, 2016 11 / 12
Intrinsic quantities Intrinsic quantities We say that a quantity is intrinsic to a surface if it can be written entirely in terms of the first fundamental form. Theorem Let x be a coordinate patch on a surface M. Then the Christoffel symbols of M with respect to x are intrinsic. More specifically ). Γ m ij = 1 2 k g km ( gik u j g ij u k + g kj u i MATH 2040 Normal and Geodesic Curvature December 20, 2016 11 / 12
Intrinsic quantities Intrinsic quantities We say that a quantity is intrinsic to a surface if it can be written entirely in terms of the first fundamental form. Theorem Let x be a coordinate patch on a surface M. Then the Christoffel symbols of M with respect to x are intrinsic. More specifically ). Theorem Γ m ij = 1 2 k g km ( gik u j g ij u k + g kj u i Let x be a coordinate patch on a surface M. If γ(s) is a unit speed curve then the geodesic curvature of γ is intrinsic. MATH 2040 Normal and Geodesic Curvature December 20, 2016 11 / 12
Intrinsic quantities An extrinsic formula for geodesic curvature Let x be a coordinate patch on a surface M. Let γ(s) be a unit speed curve on M. If θ denotes the angle between the binormal B and the normal n to the surface then κ g = κ cos θ. MATH 2040 Normal and Geodesic Curvature December 20, 2016 12 / 12