Connection and Curvature in Riemannian Geometry

In Geometry on Curved Spaces, we showed how different geometry can be when we are working on curved space instead of flat space, which we are usually more familiar with. We used the concept of a metric to express how the distance formula changes depending on where we are on this curved space. This gives us some way to “measure” the curvature of the space.

We also described the concept of parallel transport, which is in some way even more general than the metric, and can also be used to provide us with some measure of the curvature of a space. Although we can use concepts analogous to parallel transport even without the metric, if we do have a metric on the space and an expression for it, we can relate the concept of parallel transport to the metric, which is perhaps more intuitive. In this post, we formalize the concept of parallel transport by defining the Christoffel symbol and the Riemann curvature tensor, both of which we can obtain given the form of the metric. The Christoffel symbol and the Riemann curvature tensor are examples of the more general concepts of a connection and a curvature form, respectively, which need not be obtained from the metric.

Some Basics of Tensor Notation

First we establish some notation. We have already seen some tensor notation in Some Basics of (Quantum) Electrodynamics, but we explain a little bit more of that notation here, since it will be the language we will work in. Many of the ordinary vectors we are used to, such as the position, will be indexed by superscripts. We refer to these vectors as contravariant vectors. A common convention is to use Latin letters, such as i or j, as indices when we are working with space, and Greek letters, such as \mu and \nu, as indices when we are working with spacetime. Let us consider , for example, spacetime. An event in this spacetime is specified by its 4-position x^{\mu}, where x^{0}=ctx^{1}=xx^{2}=y, and x^{3}=z.

We will use the symbol g_{\mu\nu} for our metric, and we will also often express it as a matrix. For the case of flat spacetime, our metric is given by the Minkowski metric \eta_{\mu\nu}:

\displaystyle \eta_{\mu\nu}=\left(\begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1\end{array}\right)

We can use the metric to “raise” and “lower” indices. This is done by multiplying the metric and a vector, and summing over a common index (one will be a superscript and the other a subscript). We have introduced the Einstein summation convention in Some Basics of (Quantum) Electrodynamics, where repeated indices always imply summation, unless explicitly stated otherwise, and we will continue to use this convention for posts discussing differential geometry and the theory of relativity.

Here is an example of “lowering” the index of x^{\nu} in flat spacetime using the metric \eta_{\mu\nu} to obtain a new quantity x_{\mu}:

\displaystyle x_{\mu}=\eta_{\mu\nu}x^{\nu}

Explicitly, the components of the quantity x_{\mu} are given by x_{0}=-ctx_{1}=xx_{2}=y, and x_{3}=z. Note that the “time” component x_{0} has changed sign; this is because \eta_{00}=-1. A quantity such as x_{\mu}, which has a subscript index, is called a covariant vector.

In order to “raise” indices, we need the “inverse metricg^{\mu\nu}. For the Minkowski metric \eta_{\mu\nu}, the inverse metric \eta^{\mu\nu} has the exact same components as \eta_{\mu\nu}, but for more general metrics this may not be the case. The general procedure for obtaining the inverse metric is to consider the expression

\eta_{\mu\nu}\eta^{\nu\rho}=\delta_{\mu}^{\rho}

where \delta_{\mu}^{\rho} is the Kronecker delta, a quantity that can be expressed as the matrix

\displaystyle \delta_{\mu}^{\rho}=\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1\end{array}\right).

As a demonstration of what our notation can do, we recall the formula for the invariant spacetime interval:

\displaystyle (ds)^2=-(cdt)^2+(dx)^2+(dy)^2+(dz)^2

Using tensor notation combined with the Einstein summation convention, this can be written simply as

\displaystyle (ds)^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}.

The Christoffel Symbol and the Covariant Derivative

We now come back to the Christoffel symbol \Gamma^{\mu}_{\nu\lambda}. The idea behind the Christoffel symbol is that it is used to define the covariant derivative \nabla_{\nu}V^{\mu} of a vector V^{\mu}.

The covariant derivative is a very important concept in differential geometry (and not just in Riemannian geometry). When we take derivatives, we are actually comparing two vectors. To further explain what we mean, we recall that individually the components of the vectors can be thought of as functions on the space, and we recall the expression for the derivative from An Intuitive Introduction to Calculus:

\displaystyle \frac{df}{dx}=\frac{f(x+\epsilon)-f(x)}{(x+\epsilon)-(x)} when \epsilon is extremely small (essentially negligible)

More formally, we can write

\displaystyle \frac{df}{dx}=\lim_{\epsilon\to 0}\frac{f(x+\epsilon)-f(x)}{(x+\epsilon)-(x)}.

Therefore, employing the language of partial derivatives, we could have written the following partial derivative of the \mu-th component of an m-dimensional vector V^{\mu} on an m-dimensional space with respect to the coordinate x^{\nu}:

\displaystyle \frac{\partial V^{\mu}}{\partial x^{\nu}}=\lim_{\Delta x^{\nu}\to 0}\frac{V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})-V^{\mu}(x_{1},...,x^{\nu},...,x^{m})}{(x^{\nu}+\Delta x^{\nu})-(x^{\nu})}

The problem is that we are comparing vectors from different vector spaces. Recall from Vector Fields, Vector Bundles, and Fiber Bundles that we can think of a vector bundle as having a vector space for every point on the base space. The vector V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m}) belongs to the vector space on the point (x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m}), while the vector V^{\mu}(x_{1},...,x^{\nu},...,x^{m}) belongs to the vector space on the point (x_{1},...,x^{\nu},...,x^{m}). To be able to compare the two vectors we need to “transport” one to the other in the “correct” way, by which we mean parallel transport. Now we have seen in Geometry on Curved Spaces that parallel transport can have weird effects on vectors, and these weird effects are what the Christoffel symbol expresses.

Let \tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m}) denote the vector V^{\mu}(x_{1},...,x^{\nu},...,x^{m}) parallel transported from its original vector space on (x_{1},...,x^{\nu},...,x^{m}) to the vector space on (x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m}). The vector \tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m}) is given by the following expression:

\displaystyle \tilde{V}^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})=V^{\mu}(x_{1},...,x^{\nu},...,x^{m})-V^{\lambda}(x_{1},...,x^{\nu},...,x^{m})\Gamma^{\mu}_{\nu\lambda}(x_{1},...,x^{\nu},...,x^{m})\Delta x^{\nu}

Therefore the Christoffel symbol provides a “correction” for what happens when we parallel transport a vector from one point to another. This is an example of the concept of a connection, which, like the covariant derivative, is part of more general differential geometry beyond Riemannian geometry. The object that is to be parallel transported may not be a vector, for example when we have more general fiber bundles instead of vector bundles. However, in Riemannian geometry we will usually focus on vector bundles, in particular a special kind of vector bundle called the tangent bundle, which consists of the tangent vectors at a point.

Now there is more than one way to parallel transport a mathematical object, which means that there are many choices of a connection. However, in Riemannian geometry there is a special kind of connection that we will prefer. This is the connection that satisfies the following two properties:

\displaystyle \Gamma^{\mu}_{\nu\lambda}=\Gamma^{\mu}_{\lambda\nu}    (torsion-free)

\displaystyle \nabla_{\rho}g_{\mu\nu}    (metric compatibility)

The connection that satisfies these two properties is the one that can be obtained from the metric via the following formula:

\displaystyle \Gamma^{\mu}_{\nu\lambda}=\frac{1}{2}g^{\mu\sigma}(\partial_{\lambda}g_{\mu\sigma}+\partial_{\mu}g_{\sigma\lambda}-\partial_{\sigma}g_{\lambda\mu}).

The covariant derivative is then defined as

\displaystyle \nabla_{\nu}V^{\mu}=\lim_{\Delta x^{\nu}\to 0}\frac{V^{\mu}(x^{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})-\tilde{V}^{\mu}(x_{1},...,x^{\nu}+\Delta x^{\nu},...,x^{m})}{(x^{\nu}+\Delta x^{\nu})-(x^{\nu})}.

We are now comparing vectors belonging to the same vector space, and evaluating the expression above leads to the formula for the covariant derivative:

\displaystyle \nabla_{\nu}V^{\mu}=\partial_{\nu}V^{\mu}+\Gamma^{\mu}_{\nu\lambda}V^{\lambda}.

The Riemann Curvature Tensor

Next we consider the quantity known as the Riemann curvature tensor. It is once again related to parallel transport, in the following manner. Consider parallel transporting a vector V^{\sigma} through an “infinitesimal” distance specified by another vector A^{\mu}, and after that, through another infinitesimal distance specified by a yet another vector B^{\nu}. Then we go parallel transport it again in the opposite direction to A^{\mu}, then finally in the opposite direction to B^{\nu}. The path forms a parallelogram, and when the vector V^{\sigma} returns to its starting point it will then be changed by an amount \delta V^{\rho}. We can think of the Riemann curvature tensor as the quantity that relates all of these:

\displaystyle \delta V^{\rho}=R^{\rho}_{\ \sigma\mu\nu}V^{\sigma}A^{\mu}B^{\nu}.

Another way to put this is to consider taking the covariant derivative of the vector V^{\rho} along the same path as described above. The Riemann curvature tensor is then related to this quantity as follows:

\displaystyle \nabla_{\mu}\nabla_{\nu}V^{\rho}-\nabla_{\nu}\nabla_{\mu}V^{\rho}=R^{\rho}_{\ \sigma\mu\nu}V^{\sigma}.

Expanding the left hand side, and using the torsion-free property of the Christoffel symbol, we will find that

\displaystyle R^{\rho}_{\ \sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}.

For connections other than the torsion-free one that we chose, there will be another part of the expansion of the expression \nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu} called the torsion tensor. For our case, however, we need not worry about it and we can focus on the Riemann curvature tensor.

There is another quantity that can be obtained from the Riemann curvature tensor called the Ricci tensor, denoted by R_{\mu\nu}. It is given by

\displaystyle R_{\mu\nu}=R^{\lambda}_{\ \mu\lambda\nu}.

Following the Einstein summation convention, we sum over the repeated index \lambda, and therefore the resulting quantity will have only two indices instead of four. This is an example of the operation on tensors called contraction. If we raise one index using the metric and contract again, we obtain a quantity called the Ricci scalar, denoted R:

\displaystyle R=R^{\mu}_{\ \mu}

Example: The 2-Sphere

To provide an explicit example of the concepts discussed, we show their specific expressions for the case of a 2-sphere. We will only give the final results here. The explicit computations can be found among the references, but the reader may gain some practice, especially on manipulating tensors, by performing the calculations and checking only the answers here. In any case, since the metric is given, it is only a matter of substituting the relevant quantities into the formulas already given above.

We have already given the expression for the metric of the 2-sphere in Geometry on Curved Spaces. We recall that it in matrix form, it is given by (we change our notation for the radius of the 2-sphere to R_{0} to avoid confusion with the symbol for the Ricci scalar)

\displaystyle g_{mn}= \left(\begin{array}{cc}R_{0}^{2}&0\\ 0&R_{0}^{2}\text{sin}(\theta)^{2}\end{array}\right)

Individually, the components are (we will use \theta and \varphi instead of the numbers 1 and 2 for the indices)

\displaystyle g_{\theta\theta}=R_{0}^{2}

\displaystyle g_{\varphi\varphi}=R_{0}^{2}(\text{sin}(\theta))^{2}

The other components (g_{\theta\varphi} and g_{\varphi\theta}) are all equal to zero.

The Christoffel symbols are therefore given by

\displaystyle \Gamma^{\theta}_{\varphi\varphi}=-\text{sin}(\theta)\text{cos}(\theta)

\displaystyle \Gamma^{\varphi}_{\theta\varphi}=\text{cot}(\theta)

\displaystyle \Gamma^{\varphi}_{\varphi\theta}=\text{cot}(\theta)

The other components (\Gamma^{\theta}_{\theta\theta}, \Gamma^{\theta}_{\theta\varphi}, \Gamma^{\theta}_{\varphi\theta}, \Gamma^{\varphi}_{\theta\theta}, and \Gamma^{\varphi}_{\varphi\varphi}) are all equal to zero.

The components of the Riemann curvature tensor are given by

\displaystyle R^{\theta}_{\ \varphi\theta\varphi}=(\text{sin}(\theta))^{2}

\displaystyle R^{\theta}_{\ \varphi\varphi\theta}=-(\text{sin}(\theta))^{2}

\displaystyle R^{\varphi}_{\ \theta\theta\varphi}=-1

\displaystyle R^{\varphi}_{\ \theta\varphi\theta}=1

The other components (there are still twelve of them, so I won’t bother writing all their symbols down here anymore) are all equal to zero.

The components of the Ricci tensor is

\displaystyle R_{\theta\theta}=1

\displaystyle R_{\varphi\varphi}=(\text{sin}(\theta))^{2}

The other components (R_{\theta\varphi} and R_{\varphi\theta}) are all equal to zero.

Finally, the Ricci scalar is

\displaystyle R=\frac{2}{R_{0}^{2}}

We note that the larger the radius of the 2-sphere, the smaller the curvature. We can see this intuitively, for example, when it comes to the surface of our planet, which appears flat because the radius is so large. If our planet was much smaller, this would not be the case.

Bonus: The Einstein Field Equations of General Relativity

Given what we have discussed in this post, we can now write down here the expression for the Einstein field equations (also known simply as Einstein’s equations) of general relativity. It is given in terms of the Ricci tensor and the metric (of spacetime) via the following equation:

\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi}{c^{4}} GT_{\mu\nu}

where G is the gravitational constant, the same constant that appears in Newton’s law of universal gravitation (which is approximated by Einstein’s equations at certain limiting conditions), c is the speed of light in a vacuum, and T_{\mu\nu} is the energy-momentum tensor (also known as the stress-energy tensor), which gives the “density” of energy and momentum, as well as certain other related concepts, such as the pressure and shear stress. The symbol \Lambda refers to what is known as the cosmological constant, which was not there in Einstein’s original formulation but later added to support his view of an unchanging universe. Later, with the dawn of George Lemaitre’s theory of an expanding universe, later known as the Big Bang theory, the cosmological constant was abandoned. More recently, the universe was found to not only be expanding, but expanding at an accelerating rate, necessitating the return of the cosmological constant, with an interpretation in terms of the “vacuum energy”, also known as “dark energy”. Today the nature of the cosmological constant remains one of the great mysteries of modern physics.

Bonus: Connection and Curvature in Quantum Electrodynamics

The concepts of connection and curvature also appear in quantum field theory, in particular quantum electrodynamics (see Some Basics of (Quantum) Electrodynamics). It is the underlying concept in gauge theory, of which quantum electrodynamics is probably the simplest example. However, it is an example of differential geometry which does not make use of the metric. We consider a fiber bundle, where the base space is flat spacetime (also known as Minkowski spacetime), and the fiber is \text{U}(1), which is the group formed by the complex numbers with magnitude equal to 1, with law of composition given by multiplication (we can also think of this as a circle).

We want the group \text{U}(1) to act on the wave function (or field operator) \psi(x), so that the wave function has a “phase”, i.e. we have e^{i\phi(x)}\psi(x), where e^{i\phi(x)} is a complex number which depends on the location x in spacetime. Note that therefore different values of the wave function at different points in spacetime will have different values of the “phase”. In order to compare, them, we need a connection and a covariant derivative.

The connection we want is given by

\displaystyle i\frac{q}{\hbar c}A_{\mu}

where q is the charge of the electron, \hbar is the normalized Planck’s constant, c is the speed of light in a vacuum, and A_{\mu} is the four-potential of electrodynamics.

The covariant derivative (here written using the symbol D_{\mu})is

\displaystyle D_{\mu}\psi(x)=\partial_{\mu}\psi(x)+i\frac{q}{\hbar c}A_{\mu}\psi(x)

We will also have a concept analogous to the Riemann curvature tensor, called the field strength tensor, denoted F_{\mu\nu}. Of course, our “curvature” in this case is not the literal curvature of spacetime, as we have already specified that our spacetime is flat, but an abstract notion of “curvature” that specifies how the phase of our wavefunction changes as we move around the spacetime. This field strength tensor is given by the following expression:

F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}

This may be compared to the expression for the Riemann curvature tensor, where the connection is given by the Christoffel symbols. The first two terms of both expressions are very similar. The difference is that the expression for the Riemann curvature tensor has some extra terms that the expression for the field strength tensor does not have. However, a generalization of this procedure for quantum electrodynamics to groups other than \text{U}(1), called Yang-Mills theory, does feature extra terms in the expression for the field strength tensor that perhaps makes the two more similar.

The concepts we have discussed here can be used to derive the theory of quantum electrodynamics simply from requiring that the Lagrangian (from which we can obtain the equations of motion, see also Lagrangians and Hamiltonians) be invariant under \text{U}(1) transformations, i.e. even if we change the “phase” of the wave function at every point the Lagrangian remains the same. This is an example of what is known as gauge symmetry. Generalized to other groups such as \text{SU}(2) and \text{SU}(3), this is the idea behind gauge theories, which include Yang-Mills theory and leads to the standard model of particle physics.

References:

Christoffel Symbols on Wikipedia

Riemannian Curvature Tensor on Wikipedia

Einstein Field Equations on Wikipedia

Gauge Theory on Wikipedia

Riemann Tensor for Surface of a Sphere on Physics Pages

Ricci Tensor and Curvature Scalar for a Sphere on Physics Pages

Spacetime and Geometry by Sean Carroll

Geometry, Topology, and Physics by Mikio Nakahara

Introduction to Elementary Particle Physics by David J. Griffiths

Introduction to Quantum Field Theory by Michael Peskin and Daniel V. Schroeder

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Geometry on Curved Spaces

Differential geometry is the branch of mathematics used by Albert Einstein when he formulated the general theory of relativity, where gravity is the curvature of spacetime. It was originally invented by Carl Friedrich Gauss to study the curvature of hills and valleys in the Kingdom of Hanover.

From what I described, one may guess that differential geometry has something to do with curvature. The geometry we learn in high school only occurs on a flat surface. There we can put coordinates x and y and compute distances, angles, areas, and so on.

To imagine what geometry on curved spaces looks like, imagine a globe. Instead of x and y coordinates, we can use latitude and longitude. One can now see just how different geometry is on this globe. Vertical lines (the lines of constant x) on a flat surface are always the same distance apart. On a globe, the analogues of these vertical lines, the lines of constant longitude, are closer near the poles than they are near the equator.

Other weird things happen on our globe: One can have triangles with angles that sum to more than 180 degrees. Run two perpendicular line segments from the north pole to the equator. They will meet the equator at a right angle and form a triangle with three right angles for a total of 270 degrees. Also on the globe the ratio between the circumference of a circle to its diameter might no longer be equal to the number \pi.

To make things more explicit, we will introduce the concept of a metric (the word “metric” refers to a variety of mathematical concepts related to notion of distance – in this post we use it in the sense of differential geometry to refer to what is also called the metric tensor). The metric is an example of a mathematical object called a tensor, which we will not discuss much of in this post. Instead, we will think of the metric as expressing a kind of “distance formula” for our space, which may be curved. The part of differential geometry that makes use of the metric is called Riemannian geometry, named after the mathematician Bernhard Riemann, a student of Gauss who extended his results on curved spaces to higher dimensions.

We recall from From Pythagoras to Einstein several important versions of the “distance formula”, from the case of 2D space, to the case of 4D spacetime. We will focus on the simple case of 2D space in this post, since it is much easier to visualize; in fact, we have already given an example of a 2D space earlier, the globe, which we shall henceforth technically refer to as the 2-sphere. As we have learned in From Pythagoras to Einstein, a knowledge of the most simple cases can go very far toward the understanding of more complicated ones.

We will make a little change in our notation so as to stay consistent with the literature. Instead of the latitude, we will make use of the colatitude, written using the symbol \theta, and defined as the complementary angle to the latitude, i.e. the colatitude is 90 degrees minus the latitude. We will keep using the longitude, and we write it using the symbol \varphi. Note that even though we colloquially express our angles in degrees, for calculations we will always use radians, as is usual practice in mathematics and physics.

On a flat 2D space, the distance formula is given by

\displaystyle (\Delta x)^{2}+(\Delta y)^{2}=(\Delta s)^{2}.

It will be productive for us to work with extremely small quantities for now; from them we can obtain larger quantities later on using the language of calculus (see An Intuitive Introduction to Calculus). Adopting the notation of this language, we write

\displaystyle (dx)^{2}+(dy)^{2}=(ds)^{2}

We now give the distance formula for a 2-sphere:

\displaystyle R^{2}(d\theta)^{2}+R^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}

where R is the radius of the 2-sphere. This formula agrees with our intuition; the same difference in latitude and longitude result in a bigger distance for a bigger 2-sphere than for a smaller one, and the same difference in longitude results in a bigger distance for points near the equator than for points near the poles.

The idea behind the concept of the metric is that it gives how the distance formula changes depending on the coordinates. It is often written as a matrix (see Matrices) whose entries are the “coefficients” of the distance formula. Hence, for a flat 2D space it is given by

\displaystyle \left(\begin{array}{cc}1&0\\ 0&1\end{array}\right)

while for a 2-sphere it is given by

\displaystyle \left(\begin{array}{cc}R^{2}&0\\ 0&R^{2}\text{sin}(\theta)^{2}\end{array}\right).

We have seen that the metric can express how a space is curved. There are several other quantities related to the metric (and which can be derived from it), such as the Christoffel symbol and the Riemann curvature tensor, which express ideas related to curvature – however, unlike the metric which expresses curvature in terms of the distance formula, the Christoffel symbol and the Riemann curvature tensor express curvature in terms of how vectors (see Vector Fields, Vector Bundles, and Fiber Bundles) change as they move around the space.

The main equations of Einstein’s general theory of relativity, called the Einstein equations, relate the Riemann curvature tensor of 4D spacetime to the distribution of mass (or, more properly, the distribution of energy and momentum), expressed via the so-called energy-momentum tensor (also known as the stress-energy tensor).

The application of differential geometry is not limited to general relativity of course, and its objects of study are not limited to the metric. For example, in particle physics, gauge theories such as electrodynamics (see Some Basics of (Quantum) Electrodynamics) use the language of differential geometry to express forces like the electromagnetic force as a kind of “curvature”, even though a metric is not used to express this more “abstract” kind of curvature. Instead, a generalization of the concept of “parallel transport” is used. Parallel transport is the idea behind objects like the Christoffel symbol and the Riemann curvature tensor – it studies how vectors change as they move around the space. To generalize this, we replace vector bundles by more general fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles).

To give a rough idea of parallel transport, we give a simple example again in 2D space – this 2D space will be the surface of our planet. Now space itself is 3D (with time it forms a 4D spacetime). But we will ignore the up/down dimension for now and focus only on the north/south and east/west dimensions. In other words, we will imagine ourselves as 2D beings, like the characters in the novel Flatland by Edwin Abbott. The discussion below will not make references to the third up/down dimension.

Imagine that you are somewhere at the Equator, holding a spear straight in front of you, facing north. Now imagine you take a step forward with this spear. The spear will therefore remain parallel to its previous direction. You take another step, and another, walking forward (ignoring obstacles and bodies of water) until you reach the North Pole. Now at the North Pole, without turning, you take a step to the right. The spear is still parallel to its previous direction, because you did not turn. You just keep stepping to the right until you reach the Equator again. You are not at your previous location of course. To go back you need to walk backwards, which once again keeps the spear parallel to its previous direction.

When you finally come back to your starting location, you will find that you are not facing the same direction as when you first started. In fact, you (and the spear) will be facing the east, which is offset by 90 degrees clockwise from the direction you were facing at the beginning, despite the fact that you were keeping the spear parallel all the time.

This would not have happened on a flat space; this “turning” is an indicator that the space (the surface of our planet) is curved. The amount of turning depends, among other things, on the curvature of the space. Hence the idea of parallel transport gives us a way to actually measure this curvature. It is this idea, generalized to mathematical objects other than vectors, which leads to the abstract notion of curvature – it is a measure of the changes that occur in certain mathematical objects when you move around a space in a certain way, which would not have happened if you were on a flat space.

In closing, I would like to note that although differential geometry is probably most famous for its applications in physics (another interesting application in physics, by the way, is the so-called Berry’s phase in quantum mechanics), it is by no means limited to these applications alone, as already reflected in its historical origins, which barely have anything to do with physics. It has even found applications in number theory, via Arakelov theory. Still, it has an especially important role in physics, with much of modern physics written in its language, and many prospects for future theories depending on it. Whether in pure mathematics or theoretical physics, it is one of the most fruitful and active fields of research in modern times.

Bonus:

Since we have restricted ourselves to 2D spaces in this post, here is an example of a metric in 4D spacetime – this is the Schwarzschild metric, which describes the curved spacetime around objects like stars or black holes (it makes use of spherical polar coordinates):

\displaystyle \left(\begin{array}{cccc}-(1-\frac{2GM}{rc^{2}})&0&0&0\\0&(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\0&0&r^{2}&0\\ 0&0&0&r^{2}\text{sin}(\theta)^{2}\end{array}\right)

In other words, the “infinitesimal distance formula” for this curved spacetime is given by

\displaystyle -(1-\frac{2GM}{rc^{2}})(d(ct))^{2}+(1-\frac{2GM}{rc^{2}})^{-1}(dr)^{2}+r^{2}(d\theta)^{2}+r^{2}\text{sin}(\theta)^{2}(d\varphi)^{2}=(ds)^{2}

where G is the gravitational constant and M is the mass. Note also that as a matter of convention the time coordinate is “scaled” by the constant c (the speed of light in a vacuum).

References:

Differential Geometry on Wikipedia

Riemannian Geometry on Wikipedia

Metric Tensor on Wikipedia

Parallel Transport on Wikipedia

Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo

Geometry, Topology, and Physics by Mikio Nakahara