Definition 1.1 (Curve). A continuous function α : [a, b] R 3 α(t) = (x(t), y(t), z(t)), and each component is continuous.
|
|
- Dwayne Roderick Stevens
- 6 years ago
- Views:
Transcription
1 1 Space Curves Definition 1.1 (Curve). A continuous function α : [a, b] R 3 α(t) = (x(t), y(t), z(t)), and each component is continuous. is a curve Definition 1. (Differentiable Curve). α is a differentiable curve if x, y, z are differentiable. Definition 1.3 (Plane Curve). Curves α : [a, b] R are called plane curves. Definition 1.4 (Regular Curve). A curve is regular if the vector α (t) 0, that is, α (t) 0 Theorem 1.1. The arclength along a curve is s = b a α (t) Proof. α(t i ) α(t i 1 ) is a first approximation. By the mean value theorem, this is equal to α (t c ) t i t i 1. If we take the limit as (t i t i 1 ) 0, then we get b a α (t) Definition 1.5 (Unit Tangent Vector). Let α(t) be a regular curve. Then the unit tangent vector is T (t) = α (t) α (t) Definition 1.6 (Curvature). Let T (s) be the unit tangent vector parametrized by arclength. Then dt = k(s) n(s) with k(s) > 0 k(s) = dt is called the curvature, with n(s) as the unit normal vector. Definition 1.7 (Dot Product). Let u, v R 3. u v = x 1 x + y 1 y + z 1 z Lemma 1.. d ( u v) = u v + u v Corollary 1.3. Given a unit vector, ie u = 1 for t I, then d u to u(t) is orthogonal Theorem 1.4. Let α(s) be parametrized by arclength. Then α (s) = 1. Corollary 1.5. α and α are orthogonal. Definition 1.8 (Singular Points of the First Kind). Places where k(s) = 0 are singular points of the first kind. Definition 1.9 (Bi-normal Vector). The bi-normal vector b(s) is defined as b(s) = T (s) n(s) where signifies the cross product. By this definition, the binormal vector is orthogonal to both the unit tangent vector and the unit normal vector, and is itself a unit vector. Definition 1.10 (Frenet Trihedron). The Frenet Trihedron is the set of orthonormal vectors b(s), T (s), n(s). Lemma 1.6. b (s) = τ(s)n(s) Proof. b (s) = d T (s) n(s) = T (s) n(s) + T (s) n (s) T (s) = k(s)n(s), so b (s) = T (s) n (s), and so b (s) is perpendicular to both T (s) and b(s) (because b(s) is a unit vector) so b (s) = τ(s)n(s). 1
2 Definition 1.11 (Torsion). τ(s) is called the torsion of the curve. Lemma 1.7. If τ(s) 0, then α lies in a plane. Proof. First, fix the plane Π. Pick a point α(s 0 ) on the curve and let that be the origin and let b 0 be the binormal vector, which is constant because τ 0. Our curve lies in the plane Π. Let f(s) = α(s) b 0 f(0) = 0, so if the function has a zero derivative, then we are done. f (s) = α (s) b 0 + α(s) 0 = α (s) b 0, but, as α (s) = T (s), f (s) = 0. Theorem 1.8 (Frenet Relations). The Frenet Relations are 1. dt = k(s)n(s). db = τ(s)n(s) 3. dn = k(s)t (s) τ(s)b(s) Proof. The first two Frenet Relations are either previously defined or proved. As dn dn is perpendicular to n(s), it is = a 1(s)T (s) + a (s)b(s). n T = α 1 (T n) T n = a 1 T n = a 1 kn n = a 1 a 1 (s) = k(s) n b = α (b n) b n = a b n = a τn n = a a (s) = τ(s) Theorem 1.9 (Fenchel-Milner Theorem). Take a simple, closed, space curve c. c k(s) π and if k(s) = π then the curve is a circle. Also, if c is c knotted, then k(s) 4π c Theorem 1.10 (Fundamental Theorem for Curves). Given k(s) > 0 and τ(s), then!α(s) such that it has curvature k(s) and torsion τ(s), up to a rigid motion, that is, Ax + b where A is a matrix such that A t A = AA t = I, b is a vector, and x is an arbitrary point in space. Proof. We are given k(s) > 0 and τ(s), some function which is continuous. We will construct a curve α(s) such that α(s) passes through p and whose unit tangent vector points in the direction v 0. We have the Frenet Relations, and if we let W (s) = T (s) n(s), and with b(s) this we can rewrite the Frenet Relations, with an initial condition, as: dw = 0 k(s) 0 k(s) 0 τ(s) 0 τ(s) 0 W W (0) = v 0 v 0 v 0 v 0 This is a system of linear ordinary differential equations, and so a solution exists, and α(s) = s 0 T (u)du + p Now, we attempt to prove that this solution is unique.
3 We will translate the starting points of two such curves so that they coincide, rotate them so that their initial Frenet Trihedrons coincide, and use the fact that under rigid motions, k(s), τ(s) and arclength are invariant. Let M be a rigid motion such that Mx = M x 1 x x = A x 1 x x + Then, consider Mα(s) = β(s). Then, α, β have the same k(s), τ(s). β (s) = Aα (s), so β β = β = Aα Aα = A t Aα α = Iα α = α Consider (T, n, b) for α 1 and (T, n, b) for α, with T (0) = T (0), n(0) = n(0), b(0) = b(0) We claim that having the same T, n, b at s = 0 implies that they are the same for all s. Let f(s) = T T + n n + b b Note, f(0) = 0, so we must show that f(s) 0. So, if f (s) ( = 0, then f(s) = c, but as f(0) = 0, we must have c = 0. f (s) = T T, T T + n n, n n + b b, b b ) And, by the Frenet Relations, this is 0. Thus, α 1(s) = T (s) = T (s) = α (s), so, by integration, α 1 = α + c, but α 1 (0) = α (0), so α 1 = α So, now we have a global result. Theorem 1.11 (Isoperimetric Inequality). Let Γ be a simple, closed, plane curve. Let Γ enclose a region D with finite area A. Let length of Γ be denoted Γ = L. Then, 4πA L, and equality hol if and only if Γ is a circle. Lemma 1.1 (Geometric-Arithmetic Mean Inequality). ab a+b Lemma 1.13 (Cauchy-Schwartz Inequality). u, v u v Theorem 1.14 (Green s Theorem in the Plane). Given P (x, y) and Q(x, y) and a simple closed curve Γ enclosing a region D, then ( Q P dx + Qdy = x P ) dxdy y Γ Now, we are ready to prove the Isoperimetric Inequality. D Proof. Step 1 is to slide two parallel lines towar D until the fixed point of contact. Step is to set up coordinates and a circle of radius r. Step 3 is to Parametrize Γ by arclength s so Γ = (x(s), y(s)) = α(s) so α (s) = 1 The circle C is also parametrized as (x(s), y(s) Step 4 is to apply Green s Theorem A = Γ x dy = Γ x and πr = C y dx So, A + πr = C 0 ( x dy y dx ) L 0 x dy c 1 c c y dx 3
4 By the Cauchy-Schwartz Inequality, this is equal to L 0 (x + y ) 1/ ( dx + dy )1/ L = r = rl 0 By the Geometric-Arithmetic Mean Inequality,we have that Aπr A + πr rl (1), and so Aπr r L 4 4Aπ L. Assume that 4Aπ = L. Then, we need that Γ is a circle. So, we show that if 4πA = L, then r = f(l, A). From (1), we have that if L = 4Aπ, then Aπr = rl, then r = L± L 4πA And so, A + πr = L 0 π (xy yx ) = = L π L 0 ( x + y ) 1/ ( x + y ) 1/ Thus, (xy yx ) = (x + y )(x + y ) (xx + yy ) = 0 So, xx + yy = 0, then we divide by x y to get x y = y x = a. So, y = ax and x = ay. This means that y = a x and x = a y. So if we add the two equations together, we get that x + y = a (x + y ) = a Thus, x + y = ±a, meaning that a = ±r, so x y = y x = ±r So, x = ±ry, and my changing the directions of the parallel lines, we can just switch x and y, so y y 0 = ±rx. Therefore, (y y 0 ) + x = r Regular Surfaces We want to be able to do multivariable calculus on a curved surface. But first we need to discuss defining coordinates on them. Because there are no natural coordinates, we need to find the interesting geometric quantities that are coordinate invariant. One such property is Gauss Curvature. Definition.1 (Regular Surfaces). A set S in R 3 such that for every point in S we can find a ball, V, in R 3 and a map Φ from an open set U in R to V S such that 1. Φ : U V S is in C 1. Φ 1 : V S U is in C 0 3. dφ is one to one, that is, it has full rank (in the case of surfaces, rank dφ = ) It is clear that dφ = x u y u z u x v y v z v Regular Surfaces generalize to n Manifol, and are the special case where n = 4
5 Theorem.1. A surface is regular if (x u, y u, z u ) (x v, y v, z v ) 0, that is, Φ u Φ v 0 Theorem.. A graph is a regular surface, that is, z = f(x, y) y = g(x, z) and x = h(y, z) Proof. Assume the graph is of the form z = f(x, y). f C 1, so f f x and y exist and are continuous. U is the domain of f. So, let Φ = (u, v, f(u, v)). Then x(u, v) = u, y(u, v) = v and z(u, v) = f(u, v). As these are all in C 1, their partials exist, so Φ u and Φ v exist and are continuous. By simple calculation, we have Φ u Φ v 0, so z is regular. Corollary.3. The Sphere, S, is a regular surface. Consider F : W R 3 R and t = F (x, y, z) for (x, y, z) W. Definition. (Critical Point). Any point (x 0, y 0, z 0 ) with F (x 0, y 0, z 0 ) = 0 is called a critical point. Definition.3 (Critical Value). The value at a critical point is a critical value. Definition.4 (Regular Point). A noncritical point is a regular point. Definition.5 (Regular Value). The value at a regular point is a regular value. Theorem.4. Consider the level at S a = {(x, y, z) : F (x, y, z) = a}. If a is a regular value, then S a is a regular surface. Corollary.5. T, ie the two dimensional genus one torus, is a regular surface. Theorem.6 (Inverse ( Function ) Theorem). Let F : U R R : (x, y) fx f (f(x, y), g(x, y)). If y is invertible, then F has a local inverse. g x g y Theorem.7. Let S be a regular surface. Let Φ be a parametrization for p S. Φ : U V S, p V S, and with the first and third conditions of Φ verified, then if Φ 1 exists, it is continuous. Theorem.8. Given a regular surface S and any point p S, there is a neighborhood V, p V S such that the piece of the surface in V may be viewed as a graph of the type z = f(x, y) or y = g(x, z) or x = h(y, z) Proof. Our surface is regular, so we can assign local coordinates (u, v) U. Φ(u, v) = (x(u, v), y(u, v), z(u, v)) We know Φ u Φ v 0, so let us assume it is the z component that is nonzero. x x Then, u y u v y v 0 Consider Ψ : (u, v) (x(u, v), y(y, v)) with U R By the inverse function theorem, Ψ 1 exists. Definition.6 (Tangent Plane). T p S =tangent plane= {all tangent vectors of S at p} 5
6 Theorem.9. Let T be a tangent vector to S. Then, there exists an a in R such that [ x u (u x 0, v 0 ) v (u ] 0, v 0 ) y u (u y 0, v 0 ) v (u a = T 0, v 0 ) Proof. Consider (u 0, v 0 ) + t v = α(t), α (0) = v Consider Φ(α(t)) = β(t) Φ(α(0)) = Φ(u 0, v 0 ) = p β (0) = dφ(α (0)) = dφ( v) So, the set of tangent vectors T p S {dφ(p) ( ) a } b Let w T p S. w = β (0), where β(t) is a C 1 curve on S with β(0) = 0 Consider α(t) = Φ 1 (β(t)) d Φ(α(t)) = d β(t) ( a So, dφ = dφ(α b) (0)) = β (0) = w Remark - T p S is a dimensional vector space. Definition.7 (Coordinate Curves). Let p S and (u 0, v 0 ) with Φ(u 0, v 0 ) = p. Call this the center point. Consider α(t) = (u 0, v 0 + t) and β(t) = (u 0 + t, v 0 ) Then, Φ(α(t)), Φ(β(t)) are the coordinate curves through p. Remark.1. So, Φ(α(t)) = (x(u 0, v 0 + y), y(u 0, v 0 + y), z(u 0, v 0 + y)), then d Φ(α(t)) t=0 = (x v (u 0, v 0 ), y v (u 0, v 0 ), z v (u 0, v 0 )) = Φ v (u 0, v 0 ) And similarly with β and u. So, a basis for T p S is Φ v, Φ u at (u 0, v 0 ). Definition.8 (Diffeomorphism). Given regular surfaces S 1, S, assume there is a map F : S 1 S. If F is a bijection and, given p S, let u, v be local coordinates etcetera, and Ψ 1 F Φ(u, v) is differentiable and the differential is invertible, then F is a diffeomorphism. ( G1 G ) df = u u G 1 v G v So, we can say that if F is a diffeomorphism, we have df : T p (S 1 ) T F (p) (S ) Definition.9 (First Fundamental Form). Let S be a regular surface. Let E = Φ u Φu, F = Φ u Φ v, G = Φ v Φ v. Then, = E du + F du dv + G dv, and so we define = Edu + F dudv + Gdv as the first fundamental form. Definition.10 (Orthogonal Parametrization). A parametrization is orthogonal if Φ u, Φ v = 0 Definition.11 (Isothermal Coordinates). Coordinates are called isothermal if = E(u, v) ( du + dv ) Theorem.10. On a regular surface, isothermal coordinates always exist. Theorem.11. da = EG F dudv and does not depend on the coordinates. 6
7 Remark.. For a graph z f(x, y), f C 1, = (1 + f x)dx + f x f y dxdy + (1 + f y )dy =.1 Rhomb Lines, or, Loxodromes [ 1 + f ] 1/ dxdy (Theory of discontinuous groups, loxodromic/automorphic forms.) Definition.1 (Hyperbolic Metric). The metric on the Poincare half plane is = dx +dy y, and is called the hyperbolic metric. Theorem.1 (Hilbert, 1960s Efimov). If K δ < 0 an the -Manifold is complete, then one cannot embed it in R 3 Definition.13 (Rhomb Line/Loxodrome). A curve in S is called a rhomb line, or loxodrome, if and only if it intersects every meridian at a constant angle β. 3 The Gauss Map Definition 3.1 (Unit Normal of a surface). Let S be a regular surface. We define the unit normal as N(u, v) = Φu Φv Φ u Φ v Definition 3. (Gauss Map). The Gauss Map is N : S S : p N(p) and dn : T p S T N(p) S T p S Definition 3.3 (Endomorphism). A homomorphism from a group into itself is an endomorphism. Theorem 3.1. The differential of the Gauss Map is an endomorphism. Definition 3.4 (En). On the punctured sphere, the punctures are called en. Theorem 3.. The map dn : T p S T p S is a self-adjoint linear map. Proof. It is clearly linear, so all we need is dn(v), w) = v, dn(w) We have the equations and dn(φ u ), Φ u = Φ u, dn(φ u ) dn(φ v ), Φ v = Φ v, dn(φ v ) Thus, we only need to show that dn(φ u ), Φ v = Φ u, dn(φ v ), that is N u, Φ v = Φ u, N v We not that f(u, v) = N, Φ u = 0, so N v, Φ u + N, Φ uv = 0 and N u, Φ v + N, Φ uv = 0. Subtracting one from the other gives the result. [ ] a11 a Definition 3.5 (Quadratic Form). If A, is a matrix 1, then we say a 1 a Av, v = Q(v, v) is a quadratic form. Definition 3.6 (The Second Fundamental Form). II p (w) = dn(w), w is called the second fundamental form. If w = xφ u + yφ v then II p (w) = ex + fxy + gy where e = N u, Φ u, f = N u, Φ v, g = N v, Φ v. 7
8 Definition 3.7 (Normal Curvature). If w = 1, p S and w T p S, then we call II p (w) the normal curvature. So, II p (w) = k cos θ where k is the curvature of a curve on the surface and θ hte angle between N(u, v) and the normal to the curve at p. Theorem 3.3 (Meusnier). Curves passing through p and having the same tangent vector have the same normal curvature. Proof. Let α(s) be a curve α : I S in arclength parametrization, with α(0) = p, α (0) = w. Note, that N(s), α (s) = 0. Taking the derivative, we get dn, α (s) + N(s), α (s) = 0, so k cos θ = dn(s), α (s) = dn(w), w) = II p (w) [ ] [ ] cos θ x Lemma 3.4. Let w = =. Consider Q(cos θ, sin θ). sin θ y Then f(θ) = a cos θ + b cos θ sin θ + c sin θ, if max x [0,π] f(θ) = f(0) then b = 0. Proof. f(θ) is periodic, so we look at it on [ π, π]. Then f (0) = 0, so f (0) = b cos θ θ=0 = b = 0 If this is the case, then Q(x, y) = ax + cy, so f(θ) = a cos θ + c sin θ and f(0) = a. Note f(π/) = c a. Notice now that f(θ) = a cos θ + c sin θ c cos θ + c sin θ = c, so c is the minimum value, and is at a right angle to a. Application Consider w T p S. w = cos θφ u + sin θφ v. dn(w), w = e cos θ + f cos θ sin θ + g sin θ = f(θ), where e = dn(φ u ), Φ u = N u, Φ, f = N u u, Φ and g = N v v, Φ v. Assume ( that max θ [0,π] ) ( f(θ) ) = f(θ ( 0 ). Rotate ) the axes so that θ 0 = 0. cos θ0 sin θ So 0 u u = sin θ 0 cos θ 0 v v, so w = cos θ 0 Φ u +sin θ 0 Φ v. That is, w = Φ u if θ 0 = 0. dn(w), w = e, so we let w be orthogonal to w in T p S. Now we claim that w and w are principal directions. That is, dn(w) = k 1 w and dn( w) = k w. k 1, k are called the principal curvatures. Proof. Note that {w, w} is a basis for T p S. So we consider dn(w), w. Recall that w = Φ u and w = Φ v. We claim that dn(w), w = 0. b = 0 = f so f = dn(φ u ), Φ v = 0. dn(φ u ) = c 1 Φ u + c Φ v, so 0 = dn(φ u ), Φ v = c 1 Φ u, Φ v + c Φ v, Φ v. We will now check that w, w are the principal directions: Step 1: Let us call Φ u = e 1, Φ v = e. Choose the direction in the tangent plane such that the quadratic form f(θ) = e cos θ + f cos θ sin θ + g sin θ has a maximum in the direction θ = 0. Then w(θ) = e 1 cos θ + e sin θ gives dn(w), w = e cos θ + f sin θ cos θ + g sin θ. Step : At p consider w, the orthogonal direction. What is f for the pair {w, w}? f = 0 dn(φ u ), Φ v = dn(e 1 ), e = dn(w), w = 0. Step 3: Note that dn(w) = c 1 w + c w, thus 0 = dn(w), w = c 1 w, w + c w, w = c. 8
9 Definition 3.8 (Line of Curvature). A line of curvature C is a curve on S such that for all points on C, the tangent direction is principal. Theorem 3.5 (O. Rodriguez). A curve α(t) is a line of curvature iff dn = k(t)α (t), where N(t) = N(α(t)). Definition 3.9 (Gauss Curvature). The Guass curvature at p S is K = k 1 k. Definition 3.10 (Mean Curvature). The mean curvature if H = k1+k. Definition 3.11 (Classification of Points). A point p S is said to be elliptic if K(p) > 0, hyperbolic if K(p) < 0, parabolic if K(p) = 0 but either k 1 or k is not 0, and planar if k 1 = k = Euler Formula Let e 1, e be principal directions, and pick v T p S such that v = 1. Then v = cos θe 1 + sin θe. That is, {e 1, e } is an orthonormal basis of T p S. Recall: dn(v), v) =normal curvature k n = K cos φ. So we take dn(e 1 cos θ + e sin θ), e 1 cos θ + e sin θ = dn(e 1 ) cos θ + dn(e ) sin θ, e 1 cos θ + e sin θ =. k 1 cos θe 1 + k sin θe, e 1 cos θ + e sin θ = k 1 cos θ + k sin θ = k n Definition 3.1 (Umbilic Points). A point p S is said to be umbilic iff k 1 (p) = k (p) At an umbilic point, every direction is principal. Theorem 3.6. Let S be a connected surface with every point umbilic. Then S is either a piece of a sphere or a flat plane. Proof. N u = dn(φ u) = k(u, v)φ u, N v = dn(φ v) = k(u, v)φ v. Now we look at dn(φu) v dn(φv) u. So 0 = k v Φ u k u Φ v, so k v Φ u = k u Φ v. Thus, k u = k v = 0, so k = c. By connectedness, we have k = c everywhere. Now, if k = 0, then N v = N If k = c 0, then f(u, v) = Φ(u, v) + 1 c u = 0, so N(u, v) = N 0, so S is flat. f N(u, v). Thus u = Φ u + 1 N c u = 0 = 0. So f(u, v) = v 0, adn v 0 = Φ(u, v) + 1 c N v 0 and f v = Φ v + 1 N c v Φ(u, v) = 1 c N, so v 0 Φ(u, v) = 1 c, so S is a part of a sphere. Theorem 3.7 (Liebman). If S is a surface for which K(p) = c > 0, then S is a part of a sphere. Definition 3.13 (Asymptotic Direction). A direction v T p S is an Asymptotic Direction iff dn(v), v = 0. Definition 3.14 (Asymptotic Curve). A curve γ(t) S is called an asymptotic curve iff γ (t) is an asymptotic direction for all t I. Remark: If a point is elliptic, then there are no asymptotic directions. 9
10 Definition 3.15 (Minimal Surface). A Minimal Surface is a surface satisfying k 1+k = 0 for all p S. Theorem 3.8 (Osserman). If S is minimal and complete, and S \ N(S) is open, then S is the plane. Definition 3.16 (Conjugate directions). Let w 1, w T p S. We say w 1, w are conjugate directions iff dn(w 1 ), w = dn(w ), w 1 = 0. Theorem 3.9. If w is an asymptotic direction, then w is conjugate to itself. Theorem If e 1, e are principal, then they are conjugate. 3. Dupin Indicatrix Given p S, D = {w T p S : II p (w) = ±1}. Theorem D is the union of conics. Proof. Let e 1, e be principal directions, then w = r cos θe 1 + r sin θe. II p (w) = dn(w), w = r cos θk 1 e + r sin θk e, r cos θe 1 + r sin θe = r cos θk 1 + r sin θk = ±1 Case 1: k 1 > k > 0. Then r cos θk 1 + r sin θk = 1. Let ξ = r cos θ and η = r sin θ. Then ξ k 1 + η k = 1, which is the equation of an ellipse. Case : k 1 > 0 > k. Then k 1 ξ k 1 η = ±1, so we get k 1 ξ k 1 η = 1 and k 1 ξ k η = 1, a pair of hyperbolas. 3.3 Gauss Map in Local Coordinates Note: dn(φ u ) = N u = a 11Φ u +a 1 Φ v T p S, dn(φ v ) = N v = a 1Φ u +a Φ v T p S. ( ) a11 a This gives a matrix 1 with eigenvalues k a 1 a 1, k and eigenvectors the principal directsions. Note det A = k 1 k the Gauss curvature, and tr A/ = k1+k, the Mean Curvature. N u, Φ u = e = a 11 E + a 1 F N u, Φ v = f = a 11 F + a 1 G N v, Φ u = f = a 1 E + a F N v, Φ v = g = a 1 F + a G ) ( e f The above equations tell us that f g so we get the following theorem: ( a11 a = 1 a 1 a F G ) ( E F ), Theorem 3.1. ( e f f g ) ( E F F G ) 1 = ( ) a11 a 1 a 1 a 10
11 ( ) a11 a The Gauss Curvature is then K(p) = det 1. Thus K(p) = a 1 a eg f EG F. Now, recall that dn(v) = k 1 v = k 1 Iv, and dn(w) ( = k w. ) So (dn + a11 a 1, dn+ki = a 1 a ki)v = 0, and so, in the basis {Φ u, Φ v }, we have dn = ( a11 + k a 1 ), and so det(dn + ki) = (a 11 + k)(a + k) a 1 a 1 = a 1 a + k k kh + K where H is the mean curvature and K is the Gaussian curvature. Thus, k = H ± H K, and so, k 1 k k1 + k. 3.4 Equations of Weingarten ( ) 1 ( ) E F 1 G F = F G EG F F E And so, we have a 11 = a 1 = a 1 = a = ff eg EG F ef fe EG F gf fg EG F ff ge EG F Thus, H = eg ff +ge (EG F ). Theorem If (u, v) is a parametrization, f = F = 0 and p is not an umbilic point, then the coordinate curves are lines of curvature. Proof. Recall the above matrix equation. Φ(u(t), v(t)) = γ(t), γ = u Φ u + v Φ v. curvature. dn(γ ) = λ(t)γ along a line of dn(γ ) = dn(u Φ u + v Φ v ) = u dn(φ u ) + v dn(φ v ) = λu Φ u + λv Φ v = u (a 11 Φ u + a 1 Φ v ) + v (a 1 Φ u + a Φ v ), so we must have a 11 u + a 1 v = λu and a 1 u + a v = λv. ff eg That is, EG F u gf fg + EG F v = λ(t)u ef fe and EG F u ff ge + EG F v = λ(t)v are the equations of lines of curvature in general. Assume that f = F = 0, v(t) = c, then e E = λ(t). Now we assume that the coordinate curves are lines of curvature. As we are at a nonumbilic point, we have Φ u, Φ v = F = 0, and so the lines of curvature are e/e = λ 1 (t) and f/g = 0. so f = 0. Theorem A necessary and sufficient condition for coordinate curves to be asymptotic curves in a neighborhood of a hyperbolic point if e = g = 0. 11
12 Proof. II p (w) = dn(γ ), γ, γ (t) = w = u Φ u + v Φ v, so II p (w) = e(u ) + fu v + g(v ), and we need this to be 0. Assume u(t) = t and v(t) = c. Then e = g = 0. Assume e = g = 0, then we have fu v = 0. Since eg f < 0, we have f 0, so u v = 0, giving us the conclusion. Theorem 3.15 (S. Bernstein 1916). If S is a minimal surface such that it is a graph and is defined x, y R, then S is a plane. This theorem was extended in 1966 by F. Almgren and E. DiGiorgi to three dimensions, J. Simon in 1969 to n 7 and E. Bombieri, E. DiGiorgi and Ginsti proved, in 1970, that it is false for n 8. Theorem If K > 0 at p S, then the surface sits to one side of the tangent plane at p. If K < 0 at p S, then the surface sits on both sides. Proof. Φ(0, 0) = p. (Φ u (u, v) Φ(0, 0)) N at (0, 0) is ( uφu + vφ v + u Φ uu + uvφ uv + v Φ vv +... ) N = u Φ uu, N + uv Φ uv, N + v Φ vv, N = eu + fuv + gv +error It is positive at p, so all is on one side. The other part goes similarly. Theorem Let M be a compact surface, M =. Then at least one elliptic point. Remark: If M is compact and not homeomorphic to S, then M will in fact have both elliptic and hyperbolic points. Proof. Pick x 0 / M and consider a sphere S R (x 0 ). Claim: the first points touched as the sphere is contracted are elliptic. Let f(q) = q x 0 for q M. f(q) achieves maxima at x 1,.... Let α(s) M be any curve such that α(0) = x 1. Then consider g(s) = α(s) x 0 = α(s) x 0, α(s) x 0. Note g (0) = 0 and g (0) 0. g (0) = α (0), α(0) x 0 = α (0), x 1 x 0 = 0, so x 1 x 0 is normal to M at x 1. g (s) = α (s), α(s) x 0, g (s) = α (s), α(s) x 0 + α (s), α (s). g (0) = α (0), x 1 x 0 + 0, so α (0), N(0) 1. We recall that II p (α (s)) = N s, α (s), where N s = s (N(α(s))). But N s, α = N, α, so II p (x 1 ) 1, so the principal curvatures are less than zero, so elliptic. 4 Intrinsic Geometry of Surfaces Let S 1, S be surfaces, ϕ : S 1 S a diffeomorphism. Take p S 1 and a coordinate neighborhood of p. Look agt Φ 1 ϕ Φ 1 : V R W R : ( (u, v) (f(u, ) v), g(u, v)) fu f where f u, f v, g u, g v exists and are continuous, with v invertible for g u g v all (u, v) V. 1
13 Then ϕ is a diffeomorphism and if f, g C then it is a smooth diffeomorphism. Definition 4.1 (Isometry). An isometry is a map ϕ : S 1 S such that ϕ is a diffeomorphism which preserves the inner product, that is, v, w TpS 1 = dϕ(v), dϕ(w) Tϕ(p) S. Remark: if dϕ(w), dϕ(w) = w w T p S, then dϕ(w 1 ), dϕ(w ) = w 1, w. This is becuase dϕ(w) = w gives dϕ(w 1 + w ), dϕ(w 1 + w ) = w 1 + w, w 1 + w. Thus, dϕ(w 1 ), dϕ(w 1 ) + dϕ(w ), dϕ(w ) + dϕ(w 1 ), dϕ(w ) = w 1 + w + w 1, w. Definition 4. (Local Isometry). Given two regular ( surfaces ) and a map ϕ : fu f S 1 S such that ϕ is a smooth map, Jac(ϕ) = v is invertible for g u g v all (u, v) V, and p S 1, V 1 S a neighborhood of p and a corresponding neighborhood V of ϕ(p) in S such that ϕ : V 1 V is an isometry. Theorem 4.1. Let Φ : U S 1, Φ : U S. Then E = Ẽ, F = F, and G = G iff Φ 1 Φ : U U is a local isometry. Proof. Let v T p0 S 1 and w = aφ u + bφ v. Let α(t) S 1, with d t=0φ(α(t)) = w. d t=0 Φ(α(t)) = a Φ u + b Φ v = dψ(w) dψ(w) = dψ(w), dψ(w) = a Φ u + b Φ v = a ( Φ u Φu ) + ab( Φ u Φ v ) + b ( Φ v Φ v ) = a Ẽ + ab F + b G. As Ẽ = E, F = F and G = G, we have a E + abf + b G = w. The converse is trivial. Definition 4.3 (Conformal Mapping). Let ϕ : S 1 S be a diffeomorphism. If dϕ(w 1 ), dϕ(w ) Tϕ(p) S = λ (p) w 1, w TpS 1, then ϕ is conformal. Theorem 4.. Two regular surfaces are locally conformal. Theorem 4.3 (Chow). Assume H CP n a complex analytic manifold. Then H is an algebraic variety. 4.1 Christoffel Symbols Consider a regular surface S, Φ a local chart and Φ uu = aφ u + bφ v + cn = (Φ uu Φ uu, N N) + Φ uu, N N. So c = e from the second fundamental form, and a and b are called Christoffel Symbols. Definition 4.4 (Christoffel Symbols). We write Φ uu = Γ 1 11Φ u + Γ 11Φ v + en, Φ uv = Γ 1 1Φ u + Γ 1Φ v + fn, and, Φ vv = Γ 1 Φ u + Γ Φ v + gn and we call Γ k ij the Christoffel symbols. As Φ uv = Φ vu, we have Γ k ij = Γk ji, and so there are six Christoffel symbols. 13
14 Lemma 4.4. There are formulas for the Christoffel symbols in terms of E, F and G and their derivatives. Proof. By taking inner products, we get equations of the form Φ uu, Φ u = 1 E u = Γ 1 11E + Γ 11F, etcetera. By Cramer s Rule, we can solve for the Γ k ij. Now, Gauss s equation implies the Theorema Egregium, and we will also demonstrate Codazzi s Equation. Theorem 4.5 (Gauss s Equation). Assume that E 0. Then we have EK = (Γ 11) u (Γ 11) v + Γ 1 1Γ 11 + (Γ 1) Γ 11Γ Γ 11Γ 1 Proof. Observe that (Φ uu ) v = (Φ uv ) u, (Φ vv ) u = (Φ uv ) v and N uv = N vu. So we have (Φ uu ) v (Φ uv ) u = 0, we can write this as A 1 Φ u +B 1 Φ v +C 1 N = 0, and since Φ u, Φ v, N are linearly independent, A 1 = B 1 = C 1 = 0. So we start with Φ uu = Γ 1 11Φ u + Γ 11Φ v + en and Φ uv = Γ 1 1Φ u + Γ 1Φ v + fn, and so (Φ uu ) v = (Γ 1 11) u Φ u + (Γ 11) v Φ v + e v N + Γ 1 11Φ uv + Γ 11Φ vv + en v, and (Φ uv ) u = (Γ 1 1) u Φ u + (Γ 1) v Φ v + f u N + Γ 1 1Φ uu + Γ 1Φ uv + fn v. We focus on B 1, the coefficient of Φ v, and it is ( (Γ 11 ) v (Γ 1) u + Γ 1 11Γ 1 + Γ 11Γ Γ 1 1Γ 11 Γ 1Γ ) 1 Φv = 0 Corollary 4.6 (Theorema Egregium). The Gauss Curvature of a surface is invariant under a local isometry. Proposition 4.7 (Codazzi-Mainardi Equations). e v f u = eγ f(γ 1 Γ 1 11) gγ 11 and f v g u = eγ 1 + f(γ Γ 1 1) gγ 1 Theorem 4.8 (Bonnet s Theorem). Given E, F, G, e, f, g and EG F > 0 which satisfy the Gauss and Codazzi-Mainardi Equations, then there exists a map Φ : (u, v) U R R 3 such that Φ defines a regular surface with first fundamental form Edu + F dudv + Gdv and second fundamental form edu + fdudv + gdv. Additionally, this surface is unique up to rigid motion. Definition 4.5 (Orthogonal Parametrization). A parametrization is orthogonal iff F = Parallel Transport Given a tangent vector field on U S for each point, p U, there exists w(p) T p (S). Let α : [ ɛ, ɛ] U be a regular curve such that α(0) = p and α (0) = v T p S. Consider w(α(t)), and also dw/. The covariant derivative Dw (or v w, or D v w) is the orthogonal projection of dw back to the tangent plane. Note that this defines a map D v : T p S T p S Let Φ(u(t), v(t)) = α(t). Then w(u, v) = a(u, v)φ u + b(u, v)φ v, and dw = (a u u + a v v )Φ u + a(φ uu u + Φ uv v ) + (b u u + b v v )Φ v + b(φ uv u + Φ vv v ). This, after some manipulation, is equal to (a u u + a v v + aγ 1 11u + aγ 1 1v + bu Γ bv Γ 1 )Φ u + 14
15 (b u u + b v v + aγ 11u + aγ 1v + bu Γ 1 + bv Γ )Φ v + cn) The Levi-Civita Connection is our covariant derivative, and we set D Y X D X Y = [X, Y ]. So D v w doesn t depend on u or v, just the tangent directions. From this formula, all that matters is the value at zero of u, v, u, v. The curve α(t) is irrelevant, and things only depend on the first fundamental form D x i x j = Γ k ij Φ u k. D u Φ u+v Φ v w = D u Φ u w + D v Φ v w = u D Φu w + v D Φv w Definition 4.6 (Parallel). the tangent vector field w(t) is said to be parallel along α(t) iff D α (t)w = 0 Theorem 4.9. Let X(t), Y (t) be two parallel vector fiel along α(t). X(t), Y (t), X(t), Y (t) are all constants. Then d dx dy dx Proof. X(t), Y (t) =, Y + X,. As Y is tangent, we have, Y = D α X, Y and so D α X, Y + X, D α Y = d X, Y. By hypothesis, this is zero, and so X, Y is constant. Definition 4.7 (Geodesic). A curve α(t) in S is a geodesic iff D α (t)α (t) = 0. Remark 4.1. As D α (t)α (t) = 0, we have α (t) = c, and so t/c = s is the arclength, so α(t/c) = β(t) is the arc length parameterization. β (t) = α (t)/c β = 1 c α = c/c = 1 If X(0) = w and D α X = 0, then this differential equation is solved by the solution to a + au Γ 11 + av Γ bu Γ bv Γ 1 = 0 b + au Γ 11 + av Γ 1 + bu Γ 1 + bv Γ = 0 Set a = u, b = v. Then X = a(t)φ u + b(t)φ v, and we have a system of equations called the geodesic equations: u + (u ) Γ u v Γ (v ) Γ 1 = 0 v + (u ) Γ 11 + u v Γ 1 + (v ) Γ = 0 For a surface of revolution, this simplifies to u + u v f /f = 0 and v ff (u ) = 0. Theorem If Ψ is a lcoal isometry between S 1 and S, then geodesics are mapped to geodesics. Proof. Consider the geodesic equations for (u(t), v(t)). A local isometry will preserve the first fundamental form. Thus, a local isometry will preserve the geodesic equations. That is, the curve has the same image in R and the Christoffel symbols are the same. 15
16 Theorem 4.11 (Picard). Given a point p 0 S and v T p0 S, there exists a unique geodesic α(t), α(0) = p 0 and α (0) = v for t ( ɛ, ɛ). Proof. This follows from the existence and uniqueness theorem for nd order nonlinear ODEs. Theorem 4.1. If S 1, S are tangent along Γ, then covariant derivative along Γ is the same for S 1 and S. Theorem Let F : S S and assume that F is an isometry. Assume that F ix(f ) = {p F (p) = p} is a regular curve. Then F ix(f ) is a geodesic. Proof. Let p 0 F ix(f ) and γ (p 0 ) = v. By previous theorem there is a geodesic α(s) such that α(0) = p 0, α (0) = v. σ = F α is also a geodesic as F is an isometry. Moreover, σ(0) = p 0 and σ (0) = df (α (0)). So F (γ) = γ and so df (α (0)) = α (0) = v. Thus, α = σ, and so α F ix(f ), so α = γ. We are working towar the local Gauss-Bonnet Theorem, and Clairant s Theorem/Relation. 4.3 Algebraic Value of the Covariant Derivative Assume that we fix an orientation for U S, we have determined N. Let w(t) T S a tangent vector field with w = 1. We define [ ] Dw = [Dv w] = λ(t), where Dw = λ(t)n w. Note that N w is always a unit tangent vector. Definition 4.8 (Geodesic Curvature). Let [ α(s) ] be a regular curve parameterized by arclength. Geodesic curvature is k g =. Remark 4.. k g = 0 iff α(s) is a geodesic. [ Dw So ] given [ a regular curve α(t), and v(t) = w(t) = 1, v, w T S then Dv ] = dφ with φ the angle between v and w. k = kn + kg. Lemma Let α : I S be a regular curve. Let v(t), w(t) be a fmaily of unit tangent vectors along α. Then [ ] [ Dw Dv ] = dφ where φ(t) is one determination of the angle from v to w. Proof. Construct v = N(t) v(t). Now w(t) = cos φ(t)v(t) + sin φ(t) v(t). Also w (t) = sin φφ v + cos φv + cos φφ v + sin φ v. Note that w, w = 0 as w = 1. Form w = N w and [ Dw So λ(t) = dw cos φ v sin φv. So we have, N w = dw Dα ] = λ(t) where Dw = λ(t)n w., w and N w = w = cos φn v +sin φn v = sin φφ v + cos φv + cos φφ v + sin φ v, cos φ v sin φv = cos φ v, v + cos φφ + sin φφ sin φ v, v = v, V + φ And so [ ] Dw = v, v + φ = [ ] Dv + φ. 16
17 Lemma Assume (u, v) are orthogonal parameters for S. Let w(t) be a unit tangent vector field along Φ(u(t), v(t)) = α(t). Then [ ] [ ] Dw 1 = dv G u EG E du v + dφ where φ(t) is the angle between Φ u and w(t). Proof. Choose v(t) = Φ u / E. We compute dv dv, then find, N v = [ ] Dv. dv dv, N v = Φ uu du + Φ uv dv d 1 + Φ u E E = 1 E Φ uuu + Φ uv v, N Φ u = 1 E Φ uu, N Φ u u + 1 E Φ uv, N Φ v v = = G E Γ 11 Φv G, N Φ u E u + G E [ Γ 11 u + Γ 1v ] G Φv E Γ 1, N Φ u v G E And so the result hol. Definition 4.9 (Triangle). A triangle T S is a region such that 1. T is homeomorphic to a disc. T must be a piecewise regular simple closed curve having three vertices Definition 4.10 (Triangulable). A surface is said to be triangulable iff there exists a family F of triangeles {T i } such that 1. S = T i. If T i T j =, then they only share a common vertex or edge. Theorem 4.16 (Radó). A regular surface is traingulable. One can define F E + V for a triangulation to be the Euler characteristic. This is a topological invariant. Fact: In the traingulation, you can always arrange it so that each triangle is in a local chart. We define the genus g to be the number of holes or handles attached to a sphere to get the surface, and χ(s) = g. Theorem For a surface of revolution 1. Meridians are geodesics. Parallels of lattitude with f (u ) = 0 are geodesics. 3. Other geodesics intersect parallels with angel θ, and c = f(u )θ is constant. 17
18 Proof. The geodesic equations are u 1 + f f u 1u = 0 and u ff (u 1) = On a meridian, u 1 = a is constant, and u = t. So then we have a + f f a t = 0, and 0 ff 0 = 0, which are both zero, and so the meridians are geodesics.. On a parallel, u 1 = t, u = a, and so we need to have ff = 0, and f 0, so f = Let (u(t), v(t)) be a geodesic with arclength parameterization. That is, α(t) = Φ(u(t), v(t)) and α (t) = 1 = Φ u u +Φ v v. On a circle of latitude, β(t) = (f(a) cos t, f(a) sin t, g(a)) and β (t) = ( f(a) sin t, f(a) cos t, 0). So β (a) = f(a), and so cos θ = α β (t) (t) β (t). Thus, cos θ = (Φ uu + Φ v v ) (sin t, cos t, 0). So cos θ = (Φ u1 u 1 + Φ u u ) Φu 1 f(a) = Φu 1 Φu 1 f(a) u 1 = fu 1, and so cos θ = fu 1, which menas we have f cos θ = f u 1. By the geodesic equations, we have that u 1 + f (u ) f(u ) u 1u = 0, so f u 1 + f fu 1u = 0 = (f u 1). Lemma If a geodesic has been parameterized by arclength, then du 1 = cdu f(u )(f (u ) c ) 1/ Definition 4.11 (Simple Region). A simple region R S is a region R which is homeomorphic to a disc in R and R is piecewise regular and simple. Theorem 4.19 (Local Gauss-Bonnet). Consider a simple region R S. Assume a positive orientation for R = Γ 1 + Γ Γ n. Then n k g (s) + Γ i i=1 R Kdσ + n θ i = π i=1 where θ k is the exterior angle between Γ k and Γ k+1. Theorem 4.0 (Global Gauss-Bonnet). Given any region R with boundary R (which may be empty), then n k g (s) + Γ i i=1 R Kdσ + n θ i = πχ(r) i=1 Corollary 4.1. If S is a compact surface, then Kdσ = πχ(s) S We will prove the local result first, and we will need the following: Theorem 4. (Turning Tangents). (φ(ti+1 ) φ(t i )) + θ i = π Now we prove Local G-B 18
19 Proof. First we recall that we can choose [( an ) orthogonal ( ) coordinate ] system = Edu + Gdv G. So now K = EG u E + EG v 1 EG Now we assume that R is in our orthogonal coordinate chart. So then Kdσ = K(u, v) EGdudv = 1 ( ( ) Gu + ( )) Ev dudv R R R u EG v EG ( ) Then, by Green s Theorem, we have that 1 EG G u dv R + Ev du EG Recall that [ ] ( Dw = 1 dv Gu EG + E ) v du + dφ where φ is the angle from Φ u to w(s) along α(s). Apply this to w = α, then Kdσ = 1 R u ( Gu EG dv v G v du EG ) R = 1 k g (s) dφ Γi Γ i n φ n i=1 Γ i s = kdσ + k g (s) R i=1 Γ i And so φ(t i+1 ) φ(t i ) = R Kdσ + n i=1 Γ i k g (s), and so φ(t i+1 ) φ(t i ) + θ i = π = R Kσ + n i=1 k g(s) + θ i. Theorem 4.3. Let R S, and let R be bounded by C 1,..., C n piecewise regular curves. Let θ i be the external angles at the vertices v i on C i. Then n k g (s) + Kdσ + θ i = πχ(r) C i R i=1 where χ(r) = F E + V for any triangulation and C i has positive orientation. Proof. We traingulate R into T i with T i = R. For each T i we have 3 3 k g (s) + Rdσ + θ ik = π (1) k=1 C ik T i i k =1 If we sum (1) over all the triangles, we get n i=1 C i k g (s) + R Kdσ + F 3 i=1 k=1 θ ik = πf. We now must analyze F 3 i=1 k=1 θ ik. We rewrite as interior angles, θ = π φ ik with φ ik the interior angle, and so F 3 i=1 k=1 (π φ ik) = F i=1 (π 3 k=1 φ ik). This is equal to 3πF F 3 i=1 k=1 φ ik. Call E e the number of exterior edges and E I the number of interior edges, and define V e and V I similarly. Fact: 3F = E I + E e. And so 3πF = πe I + πe e, which means we have πe I + πe e F 3 i=1 k=1 φ ik = πe πe e φ ik, and as E e = V e, we have πe πv e φ ik = πe πv e πv ext φ ik. The exterior triangulation splits into two pieces. So we note that ext φ ik = vert of T i + vert of C i = πv et + π θ i = πv et +πv ec θ i = πv e θ i, and so the formula gives πe πv + θ i, and so we establish the theorem. 19
20 Corollary 4.4. If S is compact and S =, then Kdσ = πχ(s) S Corollary 4.5. A compact surface with positive curvature is homeomorphic to S. Proof. Kdσ > 0 χ(s) = S Corollary 4.6. Let K S 0, S compact and S =, then S Kdσ. 0
Final Exam Topic Outline
Math 442 - Differential Geometry of Curves and Surfaces Final Exam Topic Outline 30th November 2010 David Dumas Note: There is no guarantee that this outline is exhaustive, though I have tried to include
More informationMATH DIFFERENTIAL GEOMETRY. Contents
MATH 3968 - DIFFERENTIAL GEOMETRY ANDREW TULLOCH Contents 1. Curves in R N 2 2. General Analysis 2 3. Surfaces in R 3 3 3.1. The Gauss Bonnet Theorem 8 4. Abstract Manifolds 9 1 MATH 3968 - DIFFERENTIAL
More informationA PROOF OF THE GAUSS-BONNET THEOREM. Contents. 1. Introduction. 2. Regular Surfaces
A PROOF OF THE GAUSS-BONNET THEOREM AARON HALPER Abstract. In this paper I will provide a proof of the Gauss-Bonnet Theorem. I will start by briefly explaining regular surfaces and move on to the first
More informationDIFFERENTIAL GEOMETRY HW 4. Show that a catenoid and helicoid are locally isometric.
DIFFERENTIAL GEOMETRY HW 4 CLAY SHONKWILER Show that a catenoid and helicoid are locally isometric. 3 Proof. Let X(u, v) = (a cosh v cos u, a cosh v sin u, av) be the parametrization of the catenoid and
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More information(Differential Geometry) Lecture Notes
Evan Chen Fall 2015 This is MIT s undergraduate 18.950, instructed by Xin Zhou. The formal name for this class is Differential Geometry. The permanent URL for this document is http://web.evanchen.cc/ coursework.html,
More informationMath 5378, Differential Geometry Solutions to practice questions for Test 2
Math 5378, Differential Geometry Solutions to practice questions for Test 2. Find all possible trajectories of the vector field w(x, y) = ( y, x) on 2. Solution: A trajectory would be a curve (x(t), y(t))
More informationAn Introduction to Gaussian Geometry
Lecture Notes in Mathematics An Introduction to Gaussian Geometry Sigmundur Gudmundsson (Lund University) (version 2.068-16th of November 2017) The latest version of this document can be found at http://www.matematik.lu.se/matematiklu/personal/sigma/
More informationContents. 1. Introduction
FUNDAMENTAL THEOREM OF THE LOCAL THEORY OF CURVES KAIXIN WANG Abstract. In this expository paper, we present the fundamental theorem of the local theory of curves along with a detailed proof. We first
More informationIndex. Bertrand mate, 89 bijection, 48 bitangent, 69 Bolyai, 339 Bonnet s Formula, 283 bounded, 48
Index acceleration, 14, 76, 355 centripetal, 27 tangential, 27 algebraic geometry, vii analytic, 44 angle at a corner, 21 on a regular surface, 170 angle excess, 337 angle of parallelism, 344 angular velocity,
More information5.3 Surface Theory with Differential Forms
5.3 Surface Theory with Differential Forms 1 Differential forms on R n, Click here to see more details Differential forms provide an approach to multivariable calculus (Click here to see more detailes)
More informationLecture 13 The Fundamental Forms of a Surface
Lecture 13 The Fundamental Forms of a Surface In the following we denote by F : O R 3 a parametric surface in R 3, F(u, v) = (x(u, v), y(u, v), z(u, v)). We denote partial derivatives with respect to the
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More information1 The Differential Geometry of Surfaces
1 The Differential Geometry of Surfaces Three-dimensional objects are bounded by surfaces. This section reviews some of the basic definitions and concepts relating to the geometry of smooth surfaces. 1.1
More informationDierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algo
Dierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algorithm development Shape control and interrogation Curves
More informationSELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 2013
SELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 03 Problem (). This problem is perhaps too hard for an actual exam, but very instructional, and simpler problems using these ideas will be on the
More informationGEOMETRY HW 7 CLAY SHONKWILER
GEOMETRY HW 7 CLAY SHONKWILER 4.5.1 Let S R 3 be a regular, compact, orientable surface which is not homeomorphic to a sphere. Prove that there are points on S where the Gaussian curvature is positive,
More information9.1 Mean and Gaussian Curvatures of Surfaces
Chapter 9 Gauss Map II 9.1 Mean and Gaussian Curvatures of Surfaces in R 3 We ll assume that the curves are in R 3 unless otherwise noted. We start off by quoting the following useful theorem about self
More informationChapter 14. Basics of The Differential Geometry of Surfaces. Introduction. Parameterized Surfaces. The First... Home Page. Title Page.
Chapter 14 Basics of The Differential Geometry of Surfaces Page 649 of 681 14.1. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate
More informationDIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 5. The Second Fundamental Form of a Surface
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 5. The Second Fundamental Form of a Surface The main idea of this chapter is to try to measure to which extent a surface S is different from a plane, in other
More informationChapter 4. The First Fundamental Form (Induced Metric)
Chapter 4. The First Fundamental Form (Induced Metric) We begin with some definitions from linear algebra. Def. Let V be a vector space (over IR). A bilinear form on V is a map of the form B : V V IR which
More informationTHE FUNDAMENTAL THEOREM OF SPACE CURVES
THE FUNDAMENTAL THEOREM OF SPACE CURVES JOSHUA CRUZ Abstract. In this paper, we show that curves in R 3 can be uniquely generated by their curvature and torsion. By finding conditions that guarantee the
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationIntrinsic Surface Geometry
Chapter 7 Intrinsic Surface Geometry The second fundamental form of a regular surface M R 3 helps to describe precisely how M sits inside the Euclidean space R 3. The first fundamental form of M, on the
More informationClassical differential geometry of two-dimensional surfaces
Classical differential geometry of two-dimensional surfaces 1 Basic definitions This section gives an overview of the basic notions of differential geometry for twodimensional surfaces. It follows mainly
More informationCS Tutorial 5 - Differential Geometry I - Surfaces
236861 Numerical Geometry of Images Tutorial 5 Differential Geometry II Surfaces c 2009 Parameterized surfaces A parameterized surface X : U R 2 R 3 a differentiable map 1 X from an open set U R 2 to R
More informationGEOMETRY HW Consider the parametrized surface (Enneper s surface)
GEOMETRY HW 4 CLAY SHONKWILER 3.3.5 Consider the parametrized surface (Enneper s surface φ(u, v (x u3 3 + uv2, v v3 3 + vu2, u 2 v 2 show that (a The coefficients of the first fundamental form are E G
More informationHILBERT S THEOREM ON THE HYPERBOLIC PLANE
HILBET S THEOEM ON THE HYPEBOLIC PLANE MATTHEW D. BOWN Abstract. We recount a proof of Hilbert s result that a complete geometric surface of constant negative Gaussian curvature cannot be isometrically
More informationMath 433 Outline for Final Examination
Math 433 Outline for Final Examination Richard Koch May 3, 5 Curves From the chapter on curves, you should know. the formula for arc length of a curve;. the definition of T (s), N(s), B(s), and κ(s) for
More informationMath 426H (Differential Geometry) Final Exam April 24, 2006.
Math 426H Differential Geometry Final Exam April 24, 6. 8 8 8 6 1. Let M be a surface and let : [0, 1] M be a smooth loop. Let φ be a 1-form on M. a Suppose φ is exact i.e. φ = df for some f : M R. Show
More informationComplete Surfaces of Constant Gaussian Curvature in Euclidean Space R 3.
Summary of the Thesis in Mathematics by Valentina Monaco Complete Surfaces of Constant Gaussian Curvature in Euclidean Space R 3. Thesis Supervisor Prof. Massimiliano Pontecorvo 19 May, 2011 SUMMARY The
More informationChapter 5. The Second Fundamental Form
Chapter 5. The Second Fundamental Form Directional Derivatives in IR 3. Let f : U IR 3 IR be a smooth function defined on an open subset of IR 3. Fix p U and X T p IR 3. The directional derivative of f
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More informationM435: INTRODUCTION TO DIFFERENTIAL GEOMETRY
M435: INTODUCTION TO DIFFNTIAL GOMTY MAK POWLL Contents 7. The Gauss-Bonnet Theorem 1 7.1. Statement of local Gauss-Bonnet theorem 1 7.2. Area of geodesic triangles 2 7.3. Special case of the plane 2 7.4.
More information7.1 Tangent Planes; Differentials of Maps Between
Chapter 7 Tangent Planes Reading: Do Carmo sections 2.4 and 3.2 Today I am discussing 1. Differentials of maps between surfaces 2. Geometry of Gauss map 7.1 Tangent Planes; Differentials of Maps Between
More informationOutline of the course
School of Mathematical Sciences PURE MTH 3022 Geometry of Surfaces III, Semester 2, 20 Outline of the course Contents. Review 2. Differentiation in R n. 3 2.. Functions of class C k 4 2.2. Mean Value Theorem
More informationLinear Ordinary Differential Equations
MTH.B402; Sect. 1 20180703) 2 Linear Ordinary Differential Equations Preliminaries: Matrix Norms. Denote by M n R) the set of n n matrix with real components, which can be identified the vector space R
More informationPart IB Geometry. Theorems. Based on lectures by A. G. Kovalev Notes taken by Dexter Chua. Lent 2016
Part IB Geometry Theorems Based on lectures by A. G. Kovalev Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationDIFFERENTIAL GEOMETRY HW 5. Show that the law of cosines in spherical geometry is. cos c = cos a cos b + sin a sin b cos θ.
DIFFEENTIAL GEOMETY HW 5 CLAY SHONKWILE Show that the law of cosines in spherical geometry is 5 cos c cos a cos b + sin a sin b cos θ. Proof. Consider the spherical triangle depicted below: Form radii
More informationIntroduction to Algebraic and Geometric Topology Week 14
Introduction to Algebraic and Geometric Topology Week 14 Domingo Toledo University of Utah Fall 2016 Computations in coordinates I Recall smooth surface S = {f (x, y, z) =0} R 3, I rf 6= 0 on S, I Chart
More informationMath 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech
Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing
More informationGeometric Modelling Summer 2016
Geometric Modelling Summer 2016 Exercises Benjamin Karer M.Sc. http://gfx.uni-kl.de/~gm Benjamin Karer M.Sc. Geometric Modelling Summer 2016 1 Dierential Geometry Benjamin Karer M.Sc. Geometric Modelling
More informationDifferential Geometry II Lecture 1: Introduction and Motivation
Differential Geometry II Lecture 1: Introduction and Motivation Robert Haslhofer 1 Content of this lecture This course is on Riemannian geometry and the geometry of submanifol. The goal of this first lecture
More informationMath 497C Mar 3, Curves and Surfaces Fall 2004, PSU
Math 497C Mar 3, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 10 2.3 Meaning of Gaussian Curvature In the previous lecture we gave a formal definition for Gaussian curvature K in terms of the
More informationHOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.
HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p
More informationTotally quasi-umbilic timelike surfaces in R 1,2
Totally quasi-umbilic timelike surfaces in R 1,2 Jeanne N. Clelland, University of Colorado AMS Central Section Meeting Macalester College April 11, 2010 Definition: Three-dimensional Minkowski space R
More informationGeometry of Curves and Surfaces
Geometry of Curves and Surfaces Weiyi Zhang Mathematics Institute, University of Warwick September 18, 2014 2 Contents 1 Curves 5 1.1 Course description........................ 5 1.1.1 A bit preparation:
More informationDifferential Geometry of Surfaces
Differential Forms Dr. Gye-Seon Lee Differential Geometry of Surfaces Philipp Arras and Ingolf Bischer January 22, 2015 This article is based on [Car94, pp. 77-96]. 1 The Structure Equations of R n Definition
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More information5.2 The Levi-Civita Connection on Surfaces. 1 Parallel transport of vector fields on a surface M
5.2 The Levi-Civita Connection on Surfaces In this section, we define the parallel transport of vector fields on a surface M, and then we introduce the concept of the Levi-Civita connection, which is also
More informationν(u, v) = N(u, v) G(r(u, v)) E r(u,v) 3.
5. The Gauss Curvature Beyond doubt, the notion of Gauss curvature is of paramount importance in differential geometry. Recall two lessons we have learned so far about this notion: first, the presence
More informationSurfaces JWR. February 13, 2014
Surfaces JWR February 13, 214 These notes summarize the key points in the second chapter of Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo. I wrote them to assure that the terminology
More informationNotes on Cartan s Method of Moving Frames
Math 553 σιι June 4, 996 Notes on Cartan s Method of Moving Frames Andrejs Treibergs The method of moving frames is a very efficient way to carry out computations on surfaces Chern s Notes give an elementary
More informationDIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015
DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry
More informationPart IB GEOMETRY (Lent 2016): Example Sheet 1
Part IB GEOMETRY (Lent 2016): Example Sheet 1 (a.g.kovalev@dpmms.cam.ac.uk) 1. Suppose that H is a hyperplane in Euclidean n-space R n defined by u x = c for some unit vector u and constant c. The reflection
More informationd F = (df E 3 ) E 3. (4.1)
4. The Second Fundamental Form In the last section we developed the theory of intrinsic geometry of surfaces by considering the covariant differential d F, that is, the tangential component of df for a
More informationSOME EXERCISES IN CHARACTERISTIC CLASSES
SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationLectures 18: Gauss's Remarkable Theorem II. Table of contents
Math 348 Fall 27 Lectures 8: Gauss's Remarkable Theorem II Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams.
More informationLectures in Discrete Differential Geometry 2 Surfaces
Lectures in Discrete Differential Geometry 2 Surfaces Etienne Vouga February 4, 24 Smooth Surfaces in R 3 In this section we will review some properties of smooth surfaces R 3. We will assume that is parameterized
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationSolutions for Math 348 Assignment #4 1
Solutions for Math 348 Assignment #4 1 (1) Do the following: (a) Show that the intersection of two spheres S 1 = {(x, y, z) : (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 = r 2 1} S 2 = {(x, y, z) : (x x 2 ) 2
More informationINTRODUCTION TO GEOMETRY
INTRODUCTION TO GEOMETRY ERIKA DUNN-WEISS Abstract. This paper is an introduction to Riemannian and semi-riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames,
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More informationCourse Summary Math 211
Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.
More informationDistances, volumes, and integration
Distances, volumes, and integration Scribe: Aric Bartle 1 Local Shape of a Surface A question that we may ask ourselves is what significance does the second fundamental form play in the geometric characteristics
More information1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces
More informationChapter 16. Manifolds and Geodesics Manifold Theory. Reading: Osserman [7] Pg , 55, 63-65, Do Carmo [2] Pg ,
Chapter 16 Manifolds and Geodesics Reading: Osserman [7] Pg. 43-52, 55, 63-65, Do Carmo [2] Pg. 238-247, 325-335. 16.1 Manifold Theory Let us recall the definition of differentiable manifolds Definition
More informationMultivariable Calculus
2 Multivariable Calculus 2.1 Limits and Continuity Problem 2.1.1 (Fa94) Let the function f : R n R n satisfy the following two conditions: (i) f (K ) is compact whenever K is a compact subset of R n. (ii)
More informationGauss Theorem Egregium, Gauss-Bonnet etc. We know that for a simple closed curve in the plane. kds = 2π.
Gauss Theorem Egregium, Gauss-Bonnet etc. We know that for a simple closed curve in the plane kds = 2π. Now we want to consider a simple closed curve C in a surface S R 3. We suppose C is the boundary
More informationUSAC Colloquium. Geometry of Bending Surfaces. Andrejs Treibergs. Wednesday, November 6, Figure: Bender. University of Utah
USAC Colloquium Geometry of Bending Surfaces Andrejs Treibergs University of Utah Wednesday, November 6, 2012 Figure: Bender 2. USAC Lecture: Geometry of Bending Surfaces The URL for these Beamer Slides:
More informationGreen s Theorem in the Plane
hapter 6 Green s Theorem in the Plane Recall the following special case of a general fact proved in the previous chapter. Let be a piecewise 1 plane curve, i.e., a curve in R defined by a piecewise 1 -function
More informationChange of Variables, Parametrizations, Surface Integrals
Chapter 8 Change of Variables, Parametrizations, Surface Integrals 8. he transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it
More informationTopics. CS Advanced Computer Graphics. Differential Geometry Basics. James F. O Brien. Vector and Tensor Fields.
CS 94-3 Advanced Computer Graphics Differential Geometry Basics James F. O Brien Associate Professor U.C. Berkeley Topics Vector and Tensor Fields Divergence, curl, etc. Parametric Curves Tangents, curvature,
More informationINVERSE FUNCTION THEOREM and SURFACES IN R n
INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that
More informationHomework for Math , Spring 2008
Homework for Math 4530 1, Spring 2008 Andrejs Treibergs, Instructor April 18, 2008 Please read the relevant sections in the text Elementary Differential Geometry by Andrew Pressley, Springer, 2001. You
More informationA FEW APPLICATIONS OF DIFFERENTIAL FORMS. Contents
A FEW APPLICATIONS OF DIFFERENTIAL FORMS MATTHEW CORREIA Abstract. This paper introduces the concept of differential forms by defining the tangent space of R n at point p with equivalence classes of curves
More information(x, y) = d(x, y) = x y.
1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance
More informationTHE GAUSS MAP OF TIMELIKE SURFACES IN R n Introduction
Chin. Ann. of Math. 16B: 3(1995),361-370. THE GAUSS MAP OF TIMELIKE SURFACES IN R n 1 Hong Jianqiao* Abstract Gauss maps of oriented timelike 2-surfaces in R1 n are characterized, and it is shown that
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),
More informationDIFFERENTIAL GEOMETRY. LECTURE 12-13,
DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of
More information612 CLASS LECTURE: HYPERBOLIC GEOMETRY
612 CLASS LECTURE: HYPERBOLIC GEOMETRY JOSHUA P. BOWMAN 1. Conformal metrics As a vector space, C has a canonical norm, the same as the standard R 2 norm. Denote this dz one should think of dz as the identity
More informationThe Gauss-Bonnet Theorem
The Gauss-Bonnet Theorem Author: Andries Salm Supervisor Gil Cavalcanti July 9, 2015 1 Andries Salm The Gauss-Bonnet Theorem Special thanks go to Gil Cavalcanti, for his help and supervision. 2 Andries
More informationNotes on the Riemannian Geometry of Lie Groups
Rose- Hulman Undergraduate Mathematics Journal Notes on the Riemannian Geometry of Lie Groups Michael L. Geis a Volume, Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre
More informationHOMEWORK 3 - GEOMETRY OF CURVES AND SURFACES. where ν is the unit normal consistent with the orientation of α (right hand rule).
HOMEWORK 3 - GEOMETRY OF CURVES AND SURFACES ANDRÉ NEVES ) If α : I R 2 is a curve on the plane parametrized by arc length and θ is the angle that α makes with the x-axis, show that α t) = dθ dt ν, where
More informationGEOMETRY HW (t, 0, e 1/t2 ), t > 0 1/t2, 0), t < 0. (0, 0, 0), t = 0
GEOMETRY HW CLAY SHONKWILER Consider the map.5.0 t, 0, e /t ), t > 0 αt) = t, e /t, 0), t < 0 0, 0, 0), t = 0 a) Prove that α is a differentiable curve. Proof. If we denote αt) = xt), yt), zt0), then it
More informationStereographic projection and inverse geometry
Stereographic projection and inverse geometry The conformal property of stereographic projections can be established fairly efficiently using the concepts and methods of inverse geometry. This topic is
More informationEstimates in surfaces with positive constant Gauss curvature
Estimates in surfaces with positive constant Gauss curvature J. A. Gálvez A. Martínez Abstract We give optimal bounds of the height, curvature, area and enclosed volume of K-surfaces in R 3 bounding a
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationDifferential Geometry Exercises
Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the
More informationThe Gauss Bonnet Theorem
Chapter 27 The Gauss Bonnet Theorem That the sum of the interior angles of a triangle in the plane equals π radians was one of the first mathematical facts established by the Greeks. In 1603 Harriot 1
More information3.1 Classic Differential Geometry
Spring 2015 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry Hao Li http://cs599.hao-li.com 1 Spring 2014 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry
More informationA local characterization for constant curvature metrics in 2-dimensional Lorentz manifolds
A local characterization for constant curvature metrics in -dimensional Lorentz manifolds Ivo Terek Couto Alexandre Lymberopoulos August 9, 8 arxiv:65.7573v [math.dg] 4 May 6 Abstract In this paper we
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationVANISHING MEAN CURVATURE AND RELATED PROPERTIES
VANISHING MEAN CURVATURE AN RELATE PROPERTIES PAUL ANTONACCI 1. Introduction The project deals with surfaces that have vanishing mean curvature. A surface with vanishing mean curvature possesses interesting
More informationGaussian and Mean Curvatures
Gaussian and Mean Curvatures (Com S 477/577 Notes) Yan-Bin Jia Oct 31, 2017 We have learned that the two principal curvatures (and vectors) determine the local shape of a point on a surface. One characterizes
More information