Chern/Banchoff: Differential Geometry

2. Levi-Civita Parallelism

Levi-Civita parallelism follows as an application of the Gauss equations.

Consider a curve C on a surace X defined by uⁱ = uⁱ(t). At each point of C, let Y(t) = yⁱ(t)X_i(t) be a tangent vector to X, but not necessarily to C. Then Y'(t) = (dyⁱ/dt)X_i(t) + yⁱ(t)X_i'(t)

= (dyⁱ/dt)X_i(t) + y^j(t)[X_jk(t)du^k/dt]

= (dyⁱ/dt)X_i(t) + y^j(t)du^k/dt [_jⁱ_kX_i(t) + h_jk ]

= [dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t)] X_i(t) + y^j(t)du^k/dt h_jk.

The vector DY/dt = [dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t)] X_i(t) is the projection of the derivative of Y(t) into the tangent space at Y(t). From the vector field Y(t), we thus obtain another tangent vector field DY/dt called the absolute or covariant derivative of Y.

In differential formalism, we may write the absolute or covariant differential as DY= [dyⁱ + _jⁱ_kdu^k y^j(t)] X_i(t). This differential form depends only on the first fundamental form of X, on the equations uⁱ(t) determining the curve C, and the vector field Y(t).

Let Z(t) be a second tangential vector field along C. Then d(Y·Z) = dY · Z + Y · dZ = DY · Z + Y · DZ, since the differences DY - dY and DZ - dZ are multiples of the normal vector so their dot product with the tangent vectors Z and Y will be zero.

The vector field Y(t) is said to be parallel along C if DY = 0 for all t. It follows that if two vector fields Y and Z are parallel along C, then their dot product is constant. In particular, if Y is a parallel vector field, then the length of Y(t) is constant.

The condition that Y(t) be parallel along C is then

dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t) = 0.

This is a system of linear homogeneous ordinary differential equations of first order, with the coefficients yⁱ(t) as dependent variables and t as the independent variable. Note that the coefficients _jⁱ_kanddu^k/dt are all functions of t. By a basic existence theorem in ordinary differential equations, there exists a uniquely determined solution yⁱ(t), which takes given initial values yⁱ(t₀) at t = t₀. Geometrically this can be described by saying that a vector Y(t₀) at t₀can be displaced parallelly along C.

As an example, consider the two-dimensional Euclidean plane with first fundamental form ds² = (du¹)² +(du²)². Then _jⁱ_k = 0 and the above equations show that a vector field Y(t) is parallel if and only if dyⁱ/dt = 0 for all t, so the coefficients yⁱ(t) are constant. Thus in this case, Levi-Civita parallelism coincides with ordinary parallelism.

Consider two surfaces X and X intersecting along a curve C. The surfaces are said to be tangent along C if they have the same normal vectors at each point of C. Since the absolute differential is defined by the use of the normal vector, it follows that when two suraces are tangent to each other along C, the absolute differential is the same whether C is considered as a curve on either of the surfaces.

This property leads to the following geometrical construction of Levi-Civita parallelism The tangent planes to X along C envelop a developable surface S.

We can make this more explicit by observing that the tangent planes at X(t₀) and X(t₀+h) will intersect in a line perpendicular to both (t₀) and (t₀+h). This line has direction (t₀) x (t₀+h) = (t₀) x [(t₀+h) - (t₀)], and the limit of this as h tends to zero will have the same direction as (t₀) x '(t₀) = V(t₀). Then the surface Z(t,v) = X(t) + vV(t) will have Z₁ = X'(t) + vV'(t), Z₂= V(t), so Z₁ x Z₂ = X'(t) x V(t) + v V'(t) x V(t). But V(t) = (t) x '(t) so V'(t) = (t) x ''(t) so V'(t) lies in the tangent plane. It follows that the normal to Z is the same as the normal to X. We would like to show that the surface Z has Gaussian curvature everywhere 0. But Z₂₂ = 0 so -h₂₂ = 0 and Z₁₂ = V(t) so Z₁₂ ·(t) =-h₁₂ = 0 also. Therefore h₁₁h₂₂ - (h₁₂)² = 0 and K = 0.

To determine parallelism of a vector field Y along C is suffices to consider parallelism along C considered as a curve on S. But S is isometric to a region in the plane, where the parallelism coincides with ordinary parallelism. We may then construct parallelism on the original surface X by reversing the process.

The curve C defined by the coordinate functions uⁱ(t) is called a geodesic if it is auto-parallel , i.e. if the tangent vector field is parallel to itself along C. If X(t) gives the equation for the curve C, then the tangent vector is X'(t) = (duⁱ/dt)X_i(t). Thus in the above calculations, we use yⁱ(t) = duⁱ/dt and the condition for DX'(t)/dt = 0 is

d²(uⁱ)/dt²+ _jⁱ_kdu^k/dt du^j/dt= 0.

This is the second-order system of ordinary differential equations satisfied by the geodesics, with dependent variables uⁱ(t) and independent variable t. By a basic existence theorem in ordinary differential equations, there is a uniquely determined solution uⁱ(t) that takes on given initial values uⁱ(t₀) and duⁱ/dt at t = t₀. Geometrically, this means that for each point and each tangent vector at that point, there is a uniquely determined geodesic through that point with the given tangent vector as its derivative at the point.

Since the tangent vectors to the geodesic are parallel, in particular they have the same length. There is no loss of generality in assuming that these tangent vectors are all of unit length, so t becomes the arc-length parameter.

We now show that the geodesics on a surface can be characterized in terms of the previously defined geodesic curvature.

Proposition: The geodesics on a surface are the curves of zero geodesic curvature.

Proof: The geodesic curvature of a curve X(t) is given by k_g(t) = X''(t)·U(t)/X'(t)·X'(t), where U(t) = (t) x T(t). It can also be determined by the equation k_g(t)s'(t) = dT/dt·U(t) = DT/dt·U. Thus the condition that DT/dt = 0 implies that the geodesic curvature is zero. Moreover, since T·T = 1, we have dT/dt·T = 0 = DT/dt·T so DT/dt is a multiple of U, specifically DT/dt = k_g(t)s'(t)U(t).Thus if the geodesic curvature is zero along the curve, it follows that DT/dt = 0 so the curve is a geodesic.

Corollary: A curve C with curvature k 0 on a surface is a geodesic if and only if the principal normal is a multiple of the surface normal at every point.

Proof: By a previous exercise, k_g(t) = k(t) sin((t)) where (t) is the angle between the principal normal to the curve and the normal vector to the surface.

Exercises

On the sphere X(u¹,u²) = (asin(u¹)cos(u²), asin(u¹)sin(u²), acos(u¹)) consider the latitude circle u¹= a, constant. Show that by displacing a vector parallelly once along the circle, thw vector turns an angle of 2[pi] ( 1 - cos(a)).

Two directions V = aⁱX_iand W = b^jX_jare said to be conjugate if h_ijaⁱb^j = 0. Consider the tangent planes to a surface X along C. Show that the limit of the intersection of two neighboring tangent planes is conjugate to the tangent vector to C.

In the x-y-plane, consider the conic ax² + 2bxy + cy²= 1, ac - b²0, with center at the origin (0,0). Two lines through the origin with slopes m and m are called conjugate if a + b(m + m) + cmm = 0. Show that the midpoints of a family of parallel chords lie on a line through the origin which is conjugate to the chord of the family through the origin.

Show that the geodesic curvature of the latitude circle in Exercise 1 is cot(a) / a. Hence a latitude circle is a geodesic if and only if it is the equator.

2. Levi-Civita Parallelism Levi-Civita parallelism follows as an application of the Gauss equations. Consider a curve C on a surace X defined by uⁱ = uⁱ(t). At each point of C, let Y(t) = yⁱ(t)X_i(t) be a tangent vector to X, but not necessarily to C. Then Y'(t) = (dyⁱ/dt)X_i(t) + yⁱ(t)X_i'(t) = (dyⁱ/dt)X_i(t) + y^j(t)[X_jk(t)du^k/dt] = (dyⁱ/dt)X_i(t) + y^j(t)du^k/dt [_jⁱ_kX_i(t) + h_jk ] = [dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t)] X_i(t) + y^j(t)du^k/dt h_jk.
The vector DY/dt = [dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t)] X_i(t) is the projection of the derivative of Y(t) into the tangent space at Y(t). From the vector field Y(t), we thus obtain another tangent vector field DY/dt called the absolute or covariant derivative of Y.
In differential formalism, we may write the absolute or covariant differential as DY= [dyⁱ + _jⁱ_kdu^k y^j(t)] X_i(t). This differential form depends only on the first fundamental form of X, on the equations uⁱ(t) determining the curve C, and the vector field Y(t).
Let Z(t) be a second tangential vector field along C. Then d(Y·Z) = dY · Z + Y · dZ = DY · Z + Y · DZ, since the differences DY - dY and DZ - dZ are multiples of the normal vector so their dot product with the tangent vectors Z and Y will be zero.
The vector field Y(t) is said to be parallel along C if DY = 0 for all t. It follows that if two vector fields Y and Z are parallel along C, then their dot product is constant. In particular, if Y is a parallel vector field, then the length of Y(t) is constant.
The condition that Y(t) be parallel along C is then dyⁱ/dt + _jⁱ_kdu^k/dt y^j(t) = 0. This is a system of linear homogeneous ordinary differential equations of first order, with the coefficients yⁱ(t) as dependent variables and t as the independent variable. Note that the coefficients _jⁱ_kanddu^k/dt are all functions of t. By a basic existence theorem in ordinary differential equations, there exists a uniquely determined solution yⁱ(t), which takes given initial values yⁱ(t₀) at t = t₀. Geometrically this can be described by saying that a vector Y(t₀) at t₀can be displaced parallelly along C.
As an example, consider the two-dimensional Euclidean plane with first fundamental form ds² = (du¹)² +(du²)². Then _jⁱ_k = 0 and the above equations show that a vector field Y(t) is parallel if and only if dyⁱ/dt = 0 for all t, so the coefficients yⁱ(t) are constant. Thus in this case, Levi-Civita parallelism coincides with ordinary parallelism.
Consider two surfaces X and X intersecting along a curve C. The surfaces are said to be tangent along C if they have the same normal vectors at each point of C. Since the absolute differential is defined by the use of the normal vector, it follows that when two suraces are tangent to each other along C, the absolute differential is the same whether C is considered as a curve on either of the surfaces.
This property leads to the following geometrical construction of Levi-Civita parallelism The tangent planes to X along C envelop a developable surface S.
We can make this more explicit by observing that the tangent planes at X(t₀) and X(t₀+h) will intersect in a line perpendicular to both (t₀) and (t₀+h). This line has direction (t₀) x (t₀+h) = (t₀) x [(t₀+h) - (t₀)], and the limit of this as h tends to zero will have the same direction as (t₀) x '(t₀) = V(t₀). Then the surface Z(t,v) = X(t) + vV(t) will have Z₁ = X'(t) + vV'(t), Z₂= V(t), so Z₁ x Z₂ = X'(t) x V(t) + v V'(t) x V(t). But V(t) = (t) x '(t) so V'(t) = (t) x ''(t) so V'(t) lies in the tangent plane. It follows that the normal to Z is the same as the normal to X. We would like to show that the surface Z has Gaussian curvature everywhere 0. But Z₂₂ = 0 so -h₂₂ = 0 and Z₁₂ = V(t) so Z₁₂ ·(t) =-h₁₂ = 0 also. Therefore h₁₁h₂₂ - (h₁₂)² = 0 and K = 0.
To determine parallelism of a vector field Y along C is suffices to consider parallelism along C considered as a curve on S. But S is isometric to a region in the plane, where the parallelism coincides with ordinary parallelism. We may then construct parallelism on the original surface X by reversing the process.
The curve C defined by the coordinate functions uⁱ(t) is called a geodesic if it is auto-parallel , i.e. if the tangent vector field is parallel to itself along C. If X(t) gives the equation for the curve C, then the tangent vector is X'(t) = (duⁱ/dt)X_i(t). Thus in the above calculations, we use yⁱ(t) = duⁱ/dt and the condition for DX'(t)/dt = 0 is d²(uⁱ)/dt²+ _jⁱ_kdu^k/dt du^j/dt= 0. This is the second-order system of ordinary differential equations satisfied by the geodesics, with dependent variables uⁱ(t) and independent variable t. By a basic existence theorem in ordinary differential equations, there is a uniquely determined solution uⁱ(t) that takes on given initial values uⁱ(t₀) and duⁱ/dt at t = t₀. Geometrically, this means that for each point and each tangent vector at that point, there is a uniquely determined geodesic through that point with the given tangent vector as its derivative at the point.
Since the tangent vectors to the geodesic are parallel, in particular they have the same length. There is no loss of generality in assuming that these tangent vectors are all of unit length, so t becomes the arc-length parameter.
We now show that the geodesics on a surface can be characterized in terms of the previously defined geodesic curvature. Proposition: The geodesics on a surface are the curves of zero geodesic curvature.
Proof: The geodesic curvature of a curve X(t) is given by k_g(t) = X''(t)·U(t)/X'(t)·X'(t), where U(t) = (t) x T(t). It can also be determined by the equation k_g(t)s'(t) = dT/dt·U(t) = DT/dt·U. Thus the condition that DT/dt = 0 implies that the geodesic curvature is zero. Moreover, since T·T = 1, we have dT/dt·T = 0 = DT/dt·T so DT/dt is a multiple of U, specifically DT/dt = k_g(t)s'(t)U(t).Thus if the geodesic curvature is zero along the curve, it follows that DT/dt = 0 so the curve is a geodesic.
Corollary: A curve C with curvature k 0 on a surface is a geodesic if and only if the principal normal is a multiple of the surface normal at every point.
Proof: By a previous exercise, k_g(t) = k(t) sin((t)) where (t) is the angle between the principal normal to the curve and the normal vector to the surface.
Exercises On the sphere X(u¹,u²) = (asin(u¹)cos(u²), asin(u¹)sin(u²), acos(u¹)) consider the latitude circle u¹= a, constant. Show that by displacing a vector parallelly once along the circle, thw vector turns an angle of 2[pi] ( 1 - cos(a)).
Two directions V = aⁱX_iand W = b^jX_jare said to be conjugate if h_ijaⁱb^j = 0. Consider the tangent planes to a surface X along C. Show that the limit of the intersection of two neighboring tangent planes is conjugate to the tangent vector to C.
In the x-y-plane, consider the conic ax² + 2bxy + cy²= 1, ac - b²0, with center at the origin (0,0). Two lines through the origin with slopes m and m are called conjugate if a + b(m + m) + cmm = 0. Show that the midpoints of a family of parallel chords lie on a line through the origin which is conjugate to the chord of the family through the origin.
Show that the geodesic curvature of the latitude circle in Exercise 1 is cot(a) / a. Hence a latitude circle is a geodesic if and only if it is the equator.