The subject of Differential geometry is the application of calculus to the problem of describing curves, surfaces, and volumes in
two and three dimensions, as well as analogous structures in higher dimensions. The most immediate application of differential geometry
in geophysics is the representation of curves and surfaces in geologic models. Seismic ray tracing is an application as well, as are the
other geometric aspects of solutions to partial differential equations, such as field lines and flow lines in problems from potential theory
and from fluid dynamics.
is a running parameter along the curve and
are the unit basis vectors in the respective
The Theory of Curves
This exposition begins in 3 dimensions, but all of the results presented here generalize immediately to arbitrary dimensions.
Definition of a Curve
In (the three dimensional Euclidean space) we define a curve as a vector valued function , where
Here is a running variable along the curve. You can think of this as being like the marks on a tape measure. We assume
for now that meaning that the components of are continuous
and at least 3-times differentiable with respect to .
"In the limit as
, the vector line element defined by the difference
tends to the tangent vector at the point
The tangent vector to a curve
We can define the tangent to each point as the first derivative of with respect to
where in .
The last form with subscript is index notation.
The subtraction of the vector positions define a "directed line segment" pointing
in the direction of the lesser value of to the greater value of . As , the vector
tends to point in the direction tangent to the curve at the point .
Arclength, the natural parameter of Curves
Suppose there is a coordinate such that the tangent vector is a unit vector
Here we use the "." to indicate differentiation with respect to this special coordinate .
Alternatively, we may consider the running parameter to be a function of this new parameter such
that . This means that we can write the unit tangent vector as
From our knowledge of vectors, we may also represent the unit tangent vector as the ratio of the vector with its magnitude
, implying that which further implies
But what is ?
We note that
which implies that .
We can write formally
Formally, we may also write
meaning that is differential arc length. Thus is arc length.
The arclength is called the natural parameter of differential geometry by some authors.
"The unit tangent vector
and the unit principal normal vector
at the point
The Principal Normal vector to a curve
We can define a second vector of interest in our understanding of the applications of calculus to curves. This
is the principal normal vector.
We define the principal normal vector as the first derivative with respect to of
the unit tangent vector
But where does point?
We consider the dot product of the unit tangent vector with itself
If we differentiate both sides of the dot product with respect to
Thus, the principal normal vector is orthogonal to the tangent vector .
We can define a unit principal normal vector by dividing by its magnitude
is a circle of radius
. It's unit tangent vector is
and its unit principal normal vector is
. Here, the angle
shows the relationship between angle
in radians and arclength
Simple examples on a circle
A simple circle of radius in the provides a simple demonstration of the
tangent and principal normal vectors.
A circle may be represented as
where is the (constant) radius of the circle and is the angular coordinate. This expression
shows position on the circle of radius as a vector which points from the center of the circle to the point at
The tangent to a circle
We may differentiate the components of to obtain the tangent vector
This expression describes a tangent vector pointing in the counter-clockwise direction on the circle, where
increase in the counter-clockwise direction.
The magnitude of the tangent vector is the square root of the sum of the squares of
the components of the tangent vector
Thus, we may write the unit tangent vector as .
Thus, for a circle
which implies that
meaning that . Here is a dummy variable of integration.
If we take when . and because for a circle, we can relate to arclength via
We can rewrite the formulas for the circle and its tangent vector in term of arclenth using
The formula for the circle becomes
and the function describing its unit tangent vectors becomes
We may then take an additional derivative to obtain the principal normal vector (which is not a unit vector)
The magnitude of the principal normal vector is
We call the curvature of the circle and the radius of curvature of the circle.
Thus the unit principal normal vector is
As we can see points in a radial direction away from the center of the circle, points in the counter-clockwise tangent direction to the circle, and the principal normal vector points in the radial direction toward the center of the circle.
The Curvature vector
The notions of the tangent, principal normal, and curvature generalize beyond circles to more general curves. We need only imagine
that at each point of more general curve that there is a circle tangent to that curve that has a radius of curvature defined by the
second derivative of the function describing the curve.
We may define a curvature vector
where we define . The curvature is and the radius of curvature is . We note for future reference that
The binormal vector
You may have seen this coming. Given two unit vectors, we can define a third vector via the cross product. In this case,
we define the unit binormal vector as the cross product of the unit tangent and the unit principal normal vectors
For curves confined to a plane, this vector always points in the constant vertical direction. For more general curves, the orientation
of this vector changes as we move along the curve.
As before, differentiating the dot product with respect to
Which indicates that the derivative of the binormal vector must point orthogonal to which
is in the -plane.
We know that the binormal is perpendicular to both the tangent and the principal normal vectors. Hence and .
Differentiating with respect to arclength
points in either the or the direction
For curves not confined to a plane, it is possible for there not only to be a curvature, but also a "twist" of the curve. This twist
will be related to the derivative of the binormal vector, as this ceases to be a constant vector for curves that are not
confined to a single plane. To quantify
this twist, we define the torsion as being, from the Frenet equations,
proportional to the derivative of the binormal vector
which implies that .
Here the choice of the minus sign is a matter of convention.
To find out the value of torsion, we need a few more results. We begin with the cross product representation of the binormal vector and differentiate this with respect to arclength
We can take the dot product of both sides of this expression with
We note that and that
We can make the following replacement to obtain
Here we note that , also (because MediaWiki's math mode does not recognize the LaTeX triple dot derivative
Finally, recalling that
allowing the torsion to be written in the compact form
is a simple circular helix of radius 1. In this case, the angular coordinate is arclength
Example: a simple circular helix with a linear third coordinate
We can write the equation of a simple helix, its tangent, principal normal, and binormal vectors as
its principal normal
its third derivative
Torsion for a simple unit circular helix in arclength with a linear third coordinate
The torsion is given by the triple scalar product
Here, we note that
Hence, for the simple unit circular helix in arclength coordinates, the torsion (the "twist") is the constant factor that lifts the helix off of the plane.
The concept of torsion extends to more general curves than simple circular helixes. Just as we can imagine a circle of radius tangent to a more general plane curve, we may also consider a helix tangent to a curve that is not confined to a plane.
Helix of radius
We can generalize the example of the helix to a circular helix of radius and with angular coordinate
the tangent vectors to the helix
To calculate the unit tangent, we need the magnitude of the tangent vector
implying that for when .
Helix of radius in arclength coordinates
We can write the equation of a simple circular helix of arbitrary radius, and the associated derivative related quantities as
the helix in arclength coordinates
the unit tangent vector
the principal normal vector
the third derivative with respect to arclength
And we write the torsion as
"The unit tangent vector
, the unit principal normal vector
, and the unit binormal vector
constitute the basis of the Frenet coordinate frame. "
The Frenet coordinate frame
The unit tangent, unit principal normal, and unit binormal vectors form a coordinate frame at each point along a curve called
the Frenet frame for French astronomer and mathematician | Jean Frédéric Frenet (1816-1900) .
We recognize the following equivalent expressions which follow from even and odd permutations of the cross product
From previous results, we recall that
We note also that we can differentiate the cross product representation of
From the expression and from the definition of torsion
The Frenet equations
From the previous results we have the following autonomous system in the arclength variable of ordinary differential
equations the are, writing out the dot . derivatives as
where the last equation follows from the definition of torsion.
We may write this system in matrix and vector notation as
The Frenet result is beautiful, but suffers from the flaw that while the Frenet frame is, pointwise, an orthogonal frame, it does
not constitute an orthogonal coordinate system centered on the curve. This follows because
and do not point parallel to . In short, the principal normal
and binormal vectors are not transported in a parallel way along the curve.
The First Fundamental Form of Differential geometry
The notion of a curve can be generalized beyond orthogonal cartesian coordinates to more general coordinates. This is done by
considering differential arclength in a more general coordinate system.
We recall that the unit tangent vector is defined as the first derivative with respect to arclength of position on a curve
implying that differential position is and that differential arclength squared is given by the dot product of with
More general parameterization
Suppose that is parameterized in terms of a new coordinate in 2-dimensions, or in 3-dimensions, such
that </math>. The superscripted indices indicate that need not be an orthogonal cartesian coordinate system.
In 3-dimensions we write
and the differential position is given by
Formally, if we want the differential arclength in for this new representation, it will be
Because the dot product is on the components of the expression will be quite complicated if written out
in complete detail. Fortunately we may make use of index notation to compactify this. We may write as
Because the dot product is on the components of all possible combinations of the partial derivatives and
differentials multiply each other. In index notation this is compactly represented as
where , as well.
To further compactify the notation we define the tangents in the and directions via
Thus, we can write the differential arclength squared as
We can think of the quantity as a 2x2 matrix in 2-dimensions and a 3x3 matrix in 3-dimensions, where the elements of the matrix are dot products of tangent vectors. We define this quantity as the metric tensor . The metric tensor reduces to the identity matrix in orthogonal cartesian coordinates, and to a diagonal matrix, with values possibly other than 1 on the diagonal.
We call the expression, which is a quadratic form,
the First Fundamental Form of Differential Geometry.
Covariant and Contravariant transformations of tensors
The "covariant" or "covariance" finds several meaning in physics. The first meaning refers to a particular transformation law for changing
coordinate systems. Here, this will refer to two different possible tensor transformation
laws and the reason why we have superscripted and subscripted indexes to denote components to vectors and higher order tensors.
An easy way to see why we naturally have two different types of components of a vector can be seen in the simple example of the
full differential of a scalar function
The quantity is a scalar, possibly representing a family of level curves or surfaces. This scalar function is the result
of the inner or dot product of two vector quantities--a vector tangent to the level surfaces represented by the values of
and a differential distance along the same coordinate direction. Employing index notation, we may write
which has the form of an inner or dot product of two vectors.
Changing coordinate systems
Suppose we want to change from the to a new coordinate system coordinates.
We can transform the tangent vector
. Here .
We can transform the differentials (called the co-vector as
If we put these together, we have
. Note that the scalar function is "invariant" under the transformation from the to the .
The tangent vector follows a covariant transformation law
Were as the co-vector obeys a "contravariant transformation law"
Generalizing the co-variant and contravariant transformation laws
In general, we note that the coordinate transformation transforms
the covariant vector
Here the bar reminds us that the components of
are in the new coordinates.
The corresponding transformation for a contravariant vector may be written as
Again the bar is a reminder that the components of
and are in the new coordinates.
Definition: a contraction
A contraction is the result of summing over repeated indexes where one index is covariant and the other is contravariant. For example
the contraction of a contravariant, with a covariant is an invariant (a scalar).
, with the
for points on the surface. The coordinate system
is not, in general an orthogonal coordinate system. "
The Theory of Surfaces
A surface is structure that is a geometrical object parameterized in two dimensions via .
A family of level surfaces may be defined with an additional coordinate which is equal to a constant on a given
The parameters describe coordinate curves on the surface allowing us to write expressions for
tangent vectors to each coordinate curve
The tangent vectors span the tangent plane to any given point . Because the coordinates are not, in general orthogonal coordinates, the
tangent vectors, are not, in general orthogonal.
We recall that the components of the metric tensor are given by dot products of tangent vectors. In this
case we note for future reference that , meaning that .
"A plane spanned by (not unit) vectors
in a non-orthogonal coordinate system. There are two ways of representing a vector
in terms of components. a)
One way is by finding vectors that are parallel to the coordinate axes that sum to the vector by the parallelogram law. b)
The other way is the traditional method of projecting orthogonally via the dot product."
The Covariant and Contravariant components of a vector
Because we are no longer confined to considering only orthogonal coordinate frames, we have the possibility of representing the components
of a given vector in either covariant or contravariant coordinates. The mental picture that the reader may already have of representing
a vector by its components, would be by taking the dot product of the vector with the unit basis vectors in the given coordinate axes. Thus
we use the scalar product to project the vector down a perpendicular to the coordinate direction. This is the "normal" way to find the
components of a vector in a coordinate system.
There is another way that we can form
a basis of a vector, and that is to find the two vectors in the coordinate axis directions that sum to our vector via the parallelogram law.
For orthogonal cartesian coordinates, the are the same coordinates as from the perpendicular projection. In the case of a non-orthogonal coordinate frame, the projection is parallel to the other coordinate axis.
Suppose the vector is in a plane spanned by the two vectors that are the tangent vectors to a point on a surface . Because are not, in general, arclength coordinates on the surface, the tangent vectors are not, in general unit tangent vectors.
The Covariant components of a vector
We can easily form the covariant components of a vector by noting formally that
but we also are aware that the tangents are not unit tangents, so there is a factor of may be moved from the unit tangents to the components
We can work this from the other direction by writing the dot product of with the unit tangent vectors, respectively
. Here, the is the angle between and the respective direction.
When we compare this with the previous representation of the covariant components of
we see that the covariant components of are
The presence of the factor of reminds us that the coordinates are not arclength coordinates, otherwise, the tangent vectors would be unit tangent vectors.
"Details of the contravariant components of a vector
. The contravariant components are the parallel projections of the vectors."
The Contravariant Components of a vector
The contravariant components of a vector require a bit more work. We can begin the same way. If is repreented
in terms of components in the respective rangent directions, this must look like a dot product with the tangents
, considering the geometry as a parallel projection to the opposing coordinate axis.
To find and we consider the cross products
We note that
Applying the law of sines we note that
We may make use of the cross product relations to substitute for the sines in the law of sines to produce
Solving for and
Here, is the area of the parallelogram with sides given by the
tangent vectors and .
is the area of the parallelogram with sides formed by the tangent
vector and the vector and similarly, the quantity
is the area of the parallelogram formed by the tangent vector and the vector .
Thus, the contravariant components of may be related to the ratios of parallelogram areas
Covariant and Contravariant transformation laws
We now have defined the covariant and contravariant components of an arbitrary vector in a coordinate frame
that is not, in general, orthogonal. The task now is to verify that these representations transform according to the respective covariant
or contravariant transformation laws.
Coordinate transformation of the covariant components of a vector
Above, we found that the representation of a vector has covariant components in the nonorthogonal coordinate
. Here is the tangent to the coordinate curve in the direction.
We consider transforming the covariant components to a new coordinate system such that
and in accordance with the covariant transformation law.
Coordinate transformation of the contravariant components of a vector
Similarly, the representation of the vector has contravariant components in the nonorthogonal coordinate
system with vectors tangent to the coordinate directions given by
(indicated by the bar)
If we wanted to transform the tangent vectors into the coordinates, this would
be the same as above, as the tangent behaves as a covariant vector
In the we may write
Comparing with the first expression for the contravariant representation of we must conclude that
in accordance with the contravariant transformation law.
Relating the covariant and the contravariant components of a vector
We may find the -th component of a vector by taking the inner product with the tangent
vector to the coordinate curve in the -th direction
Here we recognize the metric tensor .
The metric tensor may be thought of having the power to lower and index
We identify a matrix with that must be invertible if coordinate transformations are to be invertible,
hence, there should be a . Formally, this inverse should have the property that
and implying that which is called the Kronecker delta.
We can find the components of by considering formally considering the inverse of a 2x2 symmetric matrix (the metric tensor is symmetric)
Given , we know that
Applying these results to the metric tensor, we define and . Thus we may write
, and .
"The tangent vectors
and the differential distances in the
directions given by
Surface area elements
In integral calculus we all learn that a change of coordinates in terms of a differential area, volume, or hypervolume will result of
increments in the transformed coordinates, multiplied by the Jacobian of the transformation. I this section we see how our tensor representation
relates to this classical representation.
We consider the tangents, and , to a point on a surface , which are not in general arclength coordinates. We further consider an area element
formed by the cross product of increments in the and . This area element
is given by finding the cross product between the increments and .
Here the superscripts and are labels rather than indexes.
We represent these differential distances by
The differential area is given by the cross product of these differential distances
We consider the area of the parallelogram spanned by the tangent vectors, represented by the cross product of the tangent vectors, where
is the angle between the vectors, and noting
Thus, the relationship between the determinant of the metric tensor and the Jacobian of the transformation from the coordinates and the coordinates is established as
Traditionally, the Jacobian of the transformation would be represented by
which is equivalent to the cross product representation above.
The unit normal vector to a surface
A surface normal is the normal direction to the tangent plane at a given point on a surface.
Hence, in light of the previous section,
"A curve parameterized by nonorthogonal surface coordinates
has a unit tangent
, unit principal normal
, unit binormal
. The tangent to the curve is also tangent to the surface, so the unit surface normal vector
lies in the plane spanned by
The unit normal vector to a curve on a surface
A curve on a surface has its unit tangent, principal normal, and binormal vectors . There is no reason that the unit normal to the surface along the curve be co-aligned with the unit principal normal or the unit binormal vectors .
What we do know is that because the tangent to a curve on a surface is also a tangent to the surface, the surface normal is orthogonal to . Thus the unit tangent vector is orthogonal to the , , and the vectors.
Let be the angle between the unit principal normal and the
unit surface normal vector , then
We recall that . It follows, therefore that
This quantity is the normal curvature" to the curve, this being the curvature of the curve in a plane passing
through the surface normal direction along the curve. Such a normal surface is called the normal section along the curve by some authors.
The Second Fundamental Form of Differential Geometry
The second derivatives of a curve are related to its curvature. The second derivatives of curves on a surface allow us to describe the
curvature of a surface.
Continuing using our notation of representing a curve on a surface. The tangent vectors
in the direction are given by
which, in arclength coordinates is
The matrix of second derivatives is given by
Allowing us to write
We note that
Subsitituting this in the expression above, we have
If we take the dot product of the surface unit normal vector with both sides of this result we have
Because is orthogonal to the tangent plane in the second term on the right yielding
Alternatively, we could take the of
We define the second fundamental tensor as
This suggests that in arclength variables
Finally, we may infer the Second Fundamental Form of Differential Geometry
Where the right hand side represents the dot product of incremental changes in position with incremental changes in the surface normal
In the light of the results for the second fundamental tensor and the second fundamental form, we can return to results derived above
for the normal curvature
Introducing another variable we may write
Where we have substituted for from the first fundamental form. Thus, the normal curvature
is the second fundamental form, divided by the first fundamental form. We can also define a normal curvature vector
Note that the numerator and denominator of the normal curvature are evaluated separately. There is no summation of repeated indexes in
the numerator and denominator.
The extrema of Normal Curvature---the Principal Curvatures and the Principal Curvature directions
We want to find the maximum and minimum curvatures at a particular point. To do this, we must take the expression for the normal
curvature, differentiate it with respect to some local coordinates and sent the result equal to zero.
If we are in the local coordinates and in the neighborhood of a given location,
the result for the normal curvature suggests the
Differentiating with respect to we have
Because the first and second fundamental tensors tend to constants at any given point on a surface, the derivatives of vanish as the coordinates and . We note also that
the derivatives and , yielding
where the index was changed to in the first term, and where
is recognized as being symmetric because and are symmetric.
Finally we may write
If we multiply by
this has the effect of raising one of the indexes of the second fundamental tensor and to make to make the metric tensor into a Kronecker delta
which is the eigenvalue problem
with being the eigenvalue. Because this is a symmetric problem, this implies that the eigenvectors are orthogonal
representing two orthogonal principal directions of curvature, with the eigenvalues
and representing to principal curvatures of the surface.
We can construct the solution to this problem with a few definitions. First, we represent the determinant of the second fundamental
tensor as . The determinant of the contravariant form of the metric tensor is . The determinant of . We also recognize that
Furthermore, we know that the determiniant of a matrix is the product of its eigenvalues so .
Writing the eigenvalue problem as the determinant
Multiplying this out, we obtain the characteristic equation of the eigenvalue problem,
Here we recognize that the last two terms are the determinant of allowing us to rewrite the characteristic
Another fact that we can make use of is that the trace of a non-singular symmetric matrix is invariant, so that
and that the determinant of the matrix is the product of its eigenvalues,
Given this information, we can construct the following equivalent forms of the characteristic equation
Gaussian Curvature, Mean Curvature
The product of the principal curvatures is called the Gaussian curvature .
The sum average of the principal curvatures is called the mean curvature .
This allows us to write an additional form of the characteristic equation of the principal curvatures
where is the Gaussian curvature and is the mean curvature.
Because is symmetric, the eigenvectors of this matrix are orthogonal. These directions are called
the principal directions.
Umbilics and elliptic, parabolic, and hyperbolic points of a surface
The sign of the curvature of a surface is determined from an examination of the unit surface normal vector
and the principal normal vector . Both of these vectors lie in the plane perpendicular to the
tangent to the curve in question. Therefore the curvature of the surface is given by
As we have seen before normal curvature is given by the ratio of the second and first fundamental forms
Here, because the first fundamental form is always positive, we see that the sign of the normal curvature is related only to
the values of the elements of the second fundamental tensor . The second fundamental form is
positive or negative definite if and only if the determinant is positive. In this
case the principal curvatures and both have the same sign, either positive of negative.
The principal directions are orthogonal, thus an inward curving (negative) or outward curving (positive) ellipsoid is described by the
principal directions. Thus, any point of a surface that has principal curvatures of the same sign is called an elliptic point.
All points of an ellipsoid are elliptic points.
If because only one principal curvature is zero, then the point is
called a parabolic point. A value of a means that the curvature is zero and that a line on which a curvature
is zero is called an asymptotic direction.
If and at a point, then the point is called a hyperbolic (or saddle) point.
If then we note from
that the second fundamental tensor is proportional to
the first fundamental tensor . Such a point is called an
umbilic. In this case, the determinants of the first and second fundamental tensors are related via .
If then the point is called an elliptic umbilic. If then
and the point is a flat spot or a parabolic umbilic.
Umbilics may be considered pathological points on a surface.
The Formulae of Weingarten
Just as for curves, the Frenet equations provide a natural coordinate frame at each point of a curve
, surfaces have a natural frame at each point.
This frame consists of the surface normal and the two tangent directions, given by . As with the Frenet frame, we can find relationships between the members of this frame and their derivatives,
though these are not, in general, arclength coordinates.
As earlier in these notes, we may differentiate the dot product of a unit vector with itself, to obtain an orthogonal vectors
While we have retained the hat symbol in the derivative, we recognize that the derivative of a unit
vector is, in general, not a unit vector.
Because the derivative of the unit surface normal points orthogonal to the tangent plane at the point where it is defined, the derivatives
must be proportional to the tangent vectors. Writing this proportionality constant by
We must find the value of . We can take the dot product of a tangent vector with each side of this
We recognize that and that
The contravariant form of the metric tensor can be used to raise the index
where we have used the fact that that .
Thus, we may write the formulae of Weingarten as
Thus, the derivative of the surface normal vector is related to the tangent vectors through the mixed form of the second fundamental
The Formulae of Gauss
For the Formulae of Gauss, we need to consider derivatives of the tangent vectors.
The second partial derivative, which is the derivative of the tangent vector must yield a result that depends both
on the tangent directions and on the normal direction. The other way of looking at this is that we need to find a way
to redefine the derivative in these non-orthogonal coordinates that acts like the derivative that we are familiar with.
The coefficients and remain to be determined.
The term with the tangent vector can be eliminated by taking the dot product of both sides with the unit surface normal vector
. Because and because
, we may immediately identify that
We may eliminate the term with by dotting both sides with a tangent vector
. Here, we recognize that
Multiplying both sides by
where and we note that .
This permits us to solve for the coefficients
The are called Christoffel symbols. Though indexed, these items do not transform
as tensors do, and are therefore symbols rather than tensors. There are variations on the notations for the Christoffel symbols.
Christoffel symbols of the first kind
With all indices covariant, we define the Christoffel symbols of the first kind as
We can immediately see that there is a symmetry of the first two indexes of the Christoffel symbol of the first kind
It is apparent that we can generate Christoffel symbols of the first kind by differentiating the covariant metric tensor
Adding and subtracting