Riemannian Geometry Zoo

Riemaniann Manifolds

The first section in our zoo is about the largest of the animals itself: the Riemannian Manifold. There are many different manifolds out there that fill an entire zoo on their own, but our zoo has one species only - the Riemannian Manifold. It belongs to the family of the differentiable manifolds and is different from the other manifolds in its family by having a metric. A metric is an animal that serves in the calculations of distances, angles, areas and volumes. Other manifolds really don’t have that and are therefore a lot more abstract then our Riemannian manifold. There are other noteworthy geometric animals in this section of the zoo, like points, curves, tangents, and continuous functions, that have a life on their own in the zoo of mathematics, but we are mostly interested in them because they also live on manifolds, and we will need them to understand the relations between the other animals in our zoo. Before we move on to the other section of the zoo, let’s have a deeper look at the manifold itself. When we dissect a smooth manifold we will find that each manifold is locally a linear space, which means that when we look close enough, no matter where, we can deal with it as if it were a flat space locally. This property is very important because it will allow us to use all kinds of linear maps on the manifold which we will visit later. A smooth manifold is a space that consists of a set of points that are connected in a smooth way. It has evolved from much simpler animals like fields C, R, or coordinate spaces \(\mathbb R^2\), … \(\mathbb R^n\). For instance a Real and smooth manifold, is in general not only described by one coordinate space, but by many - hence the name mani-fold. Let’s look at a two dimensional real and smooth manifold - for instance the surface of the sphere: The surface of the sphere cannot be represented in a continuous way by one coordinate system - simply because - no matter how we try to put one coordinate system onto the sphere - the coordinates make jumps at certain places. This doesn’t sound like a big deal - but it is - because if we want to make use of the full machinery of continuous mathematics, especially differential- and integral calculus, we need the coordinate system to be continuous - no jumps allowed. So manifolds evolved from simple coordinate spaces to a species that has many coordinate spaces that can be stitched together in a certain way. A great deal of theory went into this evolution - but we as zoologists aren’t so much concerned with this theory we can just happily accept that smooth manifolds use multiple coordinate spaces, stitched together, to keep everything smooth and hence enable us to cultivate other animals on them that previously could only live in simpler and flat spaces.

Smooth Functions and Curves

The by far larger section of our zoo has a collection of animals from the family of maps. A map is a very general concept in mathematics, but they all have in common that they take something as input and give in return something as output. The input and output might be of the same species or an entirely different animal. In our zoo we are only concerned with maps that live on Riemannian manifolds.

Let’s start by looking at one of the more familiar species, the continuous function. A continuous function will give us a scalar number at each point on the manifold.

We can think of the function as something that takes a point as input and gives a number as an output. The specialty of a continuous function is that when we move the input point continuously it will produce a continuous output. That is to say that we will not be able to move a point around and discover jumps in the output value of the function. Let’s have a closer look at this property: If we take two points close to each other the continuous function will give us two different values from the real numbers. These two values get closer to each other when we move the points closer. However, for two points, no matter how close, the function might always return two different values. Since these values are from the real numbers there are infinitely many other numbers between them. That does certainly look like a jump. Therefore the property continuity (the property of not having jumps) is precisely defined a little bit different: It is formulated like this: You give me a point and tell me a value that is so small that you would accept it as not being a jump (let’s call this value epsilon), then I can guarantee you that by moving the second point closer, that the function will return two values which are below epsilon. We call a function continuous when it fulfills this property no matter how small your epsilon is.

The idea of a moving point is a very powerful and important concept in geometry. More specifically we are interested in curves which are the trajectory of a continuously moving point. We can think of a curve as an animal that lives on the manifold. It takes a scalar parameter as input and produces a point. Continuously varying the parameter produces a continuous curve of points.

Since a curve is continuous it has a tangent at every point along the curve. The tangent is a purely geometric object that has a direction. When we change the curve the tangent might point in a different direction.

Smooth Functions and Curves - Notation

Let \(M\) be a (real and smooth) manifold. We define an infinite-dimensional vector space over \(\mathbb R\) with the underlying set of all smooth functions \(\mathcal{C}^\infty(M)\)

and with point-wise defined operations, (i.e. for any \(p\in M\), and any \(\lambda \in \mathbb R\)):

We define a smooth curve on \(M\) as a smooth map \(\gamma: \mathbb R \to M\), where \(\mathbb R\) is understood as a \(1\)-dimensional manifold.

This definition also applies to smooth maps \(I\to M\) for an open interval \(I\subseteq \mathbb R\).

A New Animal in the Zoo

So far these animals should all look quite familiar. However, the next animal in the zoo might be a little more surprising. The name of the animal should sound familiar but when we inspect it more closely it has a few surprising properties. The animal is called a tangent vector. A tangent vector lives at the tangent space - a space at a specific point of a manifold. The space is defined by means of all possible tangents to all possible curves through that point. The surprising thing might be that a tangent vector is not the same as a tangent. It is not a geometrical object but a function - a linear map - that eats a continuous function and produces a real number. A specific tangent vector animal lives at a certain point and has the curve built into it. When we give it a continuous function it will produce a number by calculating the derivative of the function along the curve. I would argue that this is a weird animal - and that it only lives through the virtue of a specific point on a specific manifold and a specific curve through that point. But we are not here to judge animals - we are here to get to get to know the zoo!

The species of tangent vectors is particularly interesting and we should have a closer look at it. We depart from our perspective as a zoologist and take the role of a functional biologist. How does the tangent vector work, specifically how does it digest a continuous function and turn it into a real number. The organ that is responsible for this function is the differential structure. The differential structure takes a point on a manifold, a curve that goes through it, and a continuous function. It combines them in such a way that the number it returns is the derivative of the function at the given point along the given curve.

If we look at it mathematically the differential structure borrows its functionality from elementary calculus - remember - we can do that because locally we can treat the manifold as a flat space.

A Word of Warning

Tangent Vectors are the building block for many advanced concepts and it is important to understand them thoroughly. Also this is a good point to become aware that for many people there is a mismatch between the intuition for tangent vectors and the mathematical formulation. It is very important to appreciate the rigorous mathematical definition and to understand why it is not defined more closely to what people often think of. The confusion probably stems from the early exposure of many students to vector calculus, where vectors are elements of a Euclidean space. Physics makes heavy use of vector calculus for the same reason as it makes use of differential geometry: The need to understand and describe change. However, the problem that arises with vector calculus is that it describes change from an exterior view of space. This exterior view comes very natural to many students and is adopted early on. The intrinsic view where we describe change on spaces that aren’t Euclidean, but can be curved, is less intuitive and somehow feels less natural. In fact we use manifolds which describe curved spaces as locally ‘flat’ spaces. It is obvious that ‘flat’ is the less complicated case and probably therefore feels more intuitive. However, Physics needs to deal with all kinds of spaces and therefore we need to choose between the higher complexity but more general notion of curved spaces (the intrinsic view), or with many more higher dimensional flat spaces (the exterior view).

The problem that arises for many students in the understanding of differential geometry is the misunderstanding of what a vector is. It is worth noting here that as such the term vector doesn’t make much sense and the only thing that can safely be said about a vector is that it is the element of a vector space. Often vector space is understood intuitively with the image of the very specific case of Euclidean space modeled with the Cartesian real coordinate space \(\mathbb R^n\). Every student is familiar with this concept and with vector space addition and scalar multiplication. While this intuitive image serves very well for many problems, it is important to understand that it is only a very special case of a vector space. When we talk about tangent vectors in differential geometry this simplified image of a ‘vector’ (the element of a Cartesian coordinate space) gets in our way to understand the more general concepts.

The reader should keep this warning in mind when trying to understand the (at first sight unintuitive) definition of the tangent vector space of a smooth manifold.

Tangent Vectors - Notation

Let \(\gamma:\mathbb R\to M\) be a smooth curve through \(p\in M\) such that \(\gamma(\lambda_0)=p\). The directional derivative operator at \(p\) along \(\gamma\) is the linear map

where \(\mathbb R\) is understood as a \(1\)-dimensional vector space over the field \(\mathbb R\) and \(f\circ\gamma\) is the point-wise function composition \(f\) after \(\gamma\).

Note that \(f\circ\gamma\) is a map \(\mathbb R\to\mathbb R\). Hence we can calculate the ‘usual’ derivative and evaluate it at \(\lambda_0\).

In differential geometry, \(X_{\gamma,p}\) is called the tangent vector to the curve \(\gamma\) at the point \(p\in M\).

Another Word of Warning

With the previous warning in mind: we should acknowledge once more at this point that the tangent vector in differential geometry is quite a bit different from what people often think of as a vector. Once again - a vector is an element of a vector space, and the vector space we are concerned with here is quite a bit different than the prime example of the Cartesian coordinate vector space. Let’s try to develop some intuition for our new ‘kind of vector’ here before we continue with the actual definition of the tangent vector space.

When we look at what will be called the tangent vector \(X_{\gamma,p}\), then we see by its very definition, it is nothing else than a linear map from all functions to \(\mathbb R\). Remember that our manifolds are constructed to be locally similar to \(\mathbb R^{dim(M)}\), but they aren’t \(\mathbb R^{dim(M)}\). So the tangent vector is quite remarkable in that it takes a function (an element of \(\mathcal{C}^\infty(M)\)) and maps it to \(\mathbb R\). The way it is doing this is by taking the derivative (the one we know from calculus, i.e., the one that is defined on \(\mathbb R\)) of the function at a certain point in the direction of a certain curve.

In other words: give me a curve that goes through a certain point on the manifold and I can tell you the derivative of any function in the direction of the curve at this point. That is precisely the job that defines a tangent vector.

Although this definition might feel superficially complicated or convoluted it fulfills two very remarkable properties:

• We can use this definition on Cartesian coordinates without loosing any results we know from ‘old’ vector calculus in Euclidean spaces. It is kind of backward compatible.
• We can use this definition on (more general, potentially curved) smooth manifolds to denote directional derivatives.

Without this definition we would simply not know how to write down derivatives on manifolds in general. Also the tangent vector is not bound to a specific function. It’s defined with respect to a point on the manifold. So we can (notationally) reuse it to take derivatives for all smooth functions.

Intuitively, \(X_{\gamma,p}\) is the velocity or the change along \(\gamma\) at \(p\). Consider the curve \(\delta(t):=\gamma(2t)\), which is the same curve but parametrised twice as fast. We have, for any \(f\in \mathcal{C}^\infty(M)\):

\(X_{\delta,p}(f) = (f\circ\delta)'(0)=2(f\circ\gamma)'(0)=2 X_{\gamma,p}(f)\) by using the chain rule. Hence \(X_{\gamma,p}\) scales like a velocity should.

Also note that so far we have identified what we will call a ‘tangent vector’, but without defining the tangent vector space it is only a name. We will now define the tangent vector space and the cotangent vector space which will allow us to talk about tangent vectors and cotangent vectors.

Tangent Space \(T_pM\) - Notation

Let \(M\) be a manifold and \(p\in M\). The tangent space \((T_pM, \oplus, \odot)\) to \(M\) at \(p\) is the vector space over \(\mathbb R\) with underlying set

Addition and scalar multiplication both defined point-wise, i.e. for any \(f\in \mathcal{C}^\infty(M)\),

Co-Vectors

The ecosystem containing curves, and smooth functions on a manifold is a particularly fruitful one. A lot of animals evolved around it.

We have already seen the animal that has a specific curve at a specific point, and when you feed it a continuous function it will give you a real number - the tangent vector.

Now let’s look at the animal that has a specific continuous function and lives at a specific point. When you feed it a tangent vector (the animal that has a curve through that point) it will give you a real number. This species is called a co-vector. It uses the same inner organ - the differential structure - to digest the input to a real number. So it is very similar to a tangent vector but we make a distinction by their function, because the two species take different inputs to produce the same output. The one animal has a curve built-in and takes a function as input, the other one has a function built-in and eats a tangent vector.

Co-ordinates - Good or Bad?

So far we haven’t said much about co-ordinates, because we kept concepts mathematical. Once you want to start computing anything on a manifold you will need to understand the relation between co-ordinates, charts, and manifolds.

The charts are the objects that will allow us to compute something on a manifold. From the charts we get co-ordinates, and a basis for the tangent vector spaces. They allow us to use real numbers for our computations. Consequently for the practitioner the charts are very important objects.

For the mathematician and the theoretical physicist who derive facts about manifolds in general, the charts are the evil objects that need to be avoided as much as possible. The reason being that showing something in a chart-free (i.e., co-ordinate-free) way means that we can avoid to write it down for every possible co-ordinate system. It simply works for any co-ordinate system.

Charts and Co-ordinates

Let \(M\) be a \(d\)-dimensional manifold. A pair \((U,x)\) where \(U\) is an open set in the topological manifold and \(x: U \to x(U) \subseteq \mathbb R^d\) is a homeomorphism (has a continuous inverse), is said to be a chart of the manifold.

The component functions of \(x: U\to x(U)\) are the maps:

for \(1\leq i\leq d\), where \(\text{proj}_i(x(p))\) is the \(i\)-th component of \(x(p)\in \mathbb R^d\). The \(x^i(p)\) are called the co-ordinates of the point \(p\in U\) with respect to the chart \((U,x)\).

Atlas

An atlas of a manifold \(M\) is a collection \(\mathscr{A}:=\{(U_\alpha,x_\alpha)\mid \alpha \in \mathcal{A}\}\) of charts such that:

Chart Transition Maps

Two charts \((U,x)\) and \((V,y)\) are said to be \(\mathcal{C}^0\)-compatible if either \(U \cap V = \emptyset\) or the map: \(y\circ x^{-1}: x(U\cap V) \to y(U\cap V)\) is continuous.

Note that \(y\circ x^{-1}\) is a map from a subset of \(\mathbb R^d\) to a subset of \(\mathbb R^d\).

Since the maps \(x\) and \(y\) are homeomorphisms, the composition map \(y \circ x^{-1}\) is also a homeomorphism and hence continuous. Therefore, any two charts on a topological manifold are \(\mathcal{C}^0\)-compatible.

The map \(y\circ x^{-1}\) (and its inverse \(x\circ y^{-1}\)) is called the chart transition map.

Multiple Charts Constitute a Manifold

At this point if it is not clear to the reader why we go through all the hassle of defining different charts for a manifold it is a good idea to pause and ponder. The main motivation for having charts is that for many interesting and very common objects we cannot establish a unique coordinate system. To describe one mathematical object (as common as the two dimensional sphere \(S^2\) for instance) we need multiple coordinate systems. In fact the meaning of the word manifold (“of many kinds”) stems from this very property that it is not necessarily possible to describe the object with one kind of map. Although, at first sight this looks like nitpicking or overcomplicated mathematical rigor, in practice it will quickly become clear that nothing can be computed without the notion of (multiple) charts and the respective co-ordinates.

In fact one of the most remarkable properties of differential geometry is to derive and describe facts about manifolds without the use of specific co-ordinates, while providing recipes how to compute using different sets of co-ordinates. So the higher-level mathematical concepts abstract away from a particular choice of co-ordinates, because there isn’t a unique choice. Formulating all proofs and facts about a particular manifold using an arbitrary choice of coordinates would be very tedious and repetitive, so facts are described with co-ordinate independent mathematical objects as far as possible and only when it comes to computing on manifolds we resort back to a specific set of coordinates.

One thing that is quite counterintuitive for a novice reader of differential geometry is: The chart maps at first sight appear to be the computationally useful object. But in fact they are merely a mathematical concept that allows us to talk about the manifold. The useful objects for computation are the points in \(x(U) \mathbb R^d\) and the chart transition maps that map from \(x(U\cap V)\) (a subset of \(\mathbb R^d\)) to \(y((U\cap V))\) (another subset of \(\mathbb R^d\)).

The most important thing to understand at this point is that \(y\circ x^{-1}\) is a map from a subset of \(\mathbb R^d\) to a subset of \(\mathbb R^d\). So this map (once composed) ‘avoids’ the manifold in a certain sense, and hence makes it possible to compute something in co-ordinates using real numbers.

Remember, the manifold itself cannot (necessarily) be described with one set of co-ordinates, and in this sense it does not have (unique) co-ordinates. However, when we need to compute something on the manifold we need real numbers and we get them through the (composition of) chart maps. More so, once we define specific charts we can derive a bunch of objects (like chart induced bases) that are useful for computations.

Chart-Induced Basis of \(T_pM\)

If \((U,x)\) is a chart on \(M\), then the co-ordinate maps \(x^a: U \to x(U) \subseteq \mathbb R\) are smooth functions on \(U\).

Since we are only dealing with finite dimensional manifolds, we can always construct a chart-induced basis of \(T_pM\) as a set (of cardinality \(dim M\)):

The \((\frac{\partial}{\partial{x}^{a}})_{p}\) are elements of \(T_pM\), hence special tangent vectors (\(X_{\gamma_{(a)},p}\)) that are defined for a specific chart at \(p\), by defining the curves \(\gamma_{(a)}\) in a chart dependent way:

where the \((0,..,0,t,0,..,0)_{a}\) is an element of \(\mathbb R^d\), that has the \(t\) in the \(a\)-th position.

The basis vectors (being elements of \(T_pM\)) provide linear mappings:

Since it can be shown that they form a basis for our vector space \(T_pM\) it is sufficient to use linear combinations of them to act on arbitrary functions of \(\mathcal{C}^\infty(M)\). In other words: for a given chart (induced basis), all possible tangent vectors (elements of \(T_pM\)), can be written as linear combinations of the basis vectors.

The Symbol \(\partial\) Looks like a Partial Derivative - Is It?

The above definitions are fundamentally different from what you might be familiar with, from vector calculus in Euclidean spaces. In \(d\)-dimensional Euclidean space we have a set of coordinate functions that are valid for all of \(\mathbb R^d\), which gives us \(d\) many \(d\)-dimensional basis vectors. For our smooth manifold we define the basis of the tangent vector space for a specific point \(p\) and a given chart.

Also it is worth noting here that \((\frac{\partial}{\partial{x}^{a}})_{p}\) is to be read as a symbol denoting one specific element of the tangent vector space at \(p\). Specifically for this element we:

• know how to construct it from the given co-ordinate map \(x^a\) and
• intend to use it as a basis vector of \(TpM\)

The symbol resembles the partial derivative symbol and in fact for the manifold \(\mathbb R^d\), with the identity chart map, the basis vectors of \(TpM\) behave consistently with the partial derivative. Both (the partial derivative and the tangent space vector) take a function and produce the same real number. So the notation \((\frac{\partial}{\partial{x}^{a}})_{p}\) was clearly chosen to resemble the partial derivative, but it is important to understand that the partial derivative is only defined on \(\mathbb R^d\), while our tangent vectors will allow us to compute directional derivatives on smooth manifolds.

Change of Components of Chart-Induced Basis of \(T_pM\)

To change from components \(\dot{\gamma}^{j}_{x}\) of a tangent vector given in the chart (U,x)-induced basis, to components of a tangent vector in the chart (V,y)-induced basis, we need to first derive the objects \((\frac{\partial y^i}{\partial x^j})_p\)

The change of components of the tangent vector is then given as:

which is commonly written in \(\sum\)instein notation as

Gradient and Gradient Operator

The gradient of a function in \(\mathcal{C}^\infty(M)\) is a special case of the derivative. Let’s first define the gradient and the related gradient operator, which we will use to construct the Cotangent Space.

Let \(M\) be a manifold and let \(f: M \to \mathbb R\) be smooth. The gradient of \(f\) at \(p\in M\) is the covector (an element of the Cotangent space at \(p\))

In fact, we can define the gradient operator at \(p\in M\) as the \(\mathbb R\)-linear map

with \(p\in U\subseteq M\).

Cotangent Space \(T^*_pM\) and its Chart Induced Basis

Let \(M\) be a manifold and \(p\in M\) a point on the manifold and let \(T_pM\) denote the tangent space at \(p\). The cotangent space to \(M\) at \(p\) is defined as the vector space dual:

which is the set of all linear maps from elements of the tangent vector space \(T_pM\) to \(\mathbb R\).

Let \((U,x)\) be a chart on \(M\), with \(p\in U\).

We can apply the gradient operator \(d_p\) (with \(p\in U\)) to each of the basis vectors of \(\{(\frac{\partial}{\partial{x}^{a}})_{p}\}\) to obtain \((\dim M)\)-many elements of \(T^*_p M\).

The set \(\{d_px^a\mid 1\leq a \leq \dim M\}\) can be shown to form a basis of \(T^*_p M\).

If \(\{(\frac{\partial}{\partial{x}^{a}})_{p}\}\) is the basis of \(T_pM\) induced by a chart \((U,x)\), then the dual basis is denoted as \(\{({\rm d} x^a)_p\}\) and by definition (of the dual) we have

Why didn’t we need a cotangent space in Euclidean geometry?

todo

Derivative and Push-Forward

We have already seen a special case of the derivative - the gradient. Now we will define the general derivative which is also called the push-forward.

Let \(M\) and \(N\) be manifolds and let \(\phi: M \to N\) be smooth.

The differential or derivative of \(\phi\) at \(p\in M\) is the linear map

where \(d_p\phi(X)\) is the tangent vector to \(N\) at \(\phi(p)\)

todo…

The Meaning of the Co-Vector Basis

https://www.youtube.com/watch?v=CliW7kSxxWU&feature=youtu.be

Tensor Spaces

todo

The Metric Tensor

TODO

Intrinsic vs. Extrinsic

When reading differential geometry literature a lot of discussion is about the words intrinsic vs. extrinsic. Not only is the discussion confusing, even worse, the mathematical notation for intrinsic and extrinsic objects is often used as if they were interchangeable. For instance the symbols used for a tangent vector basis are often the same regardless of the basis vectors being intrinsic or extrinsic. While this approach emphasises that fundamentally they do describe the same object, it becomes very hard to see which vectors ‘live’ on the manifold (intrinsic) and which vectors ‘live’ in the ambient space.

There’s nothing one can do to clear up this confusion in literature. In this primer we exclusively will use the intrinsic viewpoint. If we need to make an exception we will clearly state that an object is ‘living’ in the ambient space and will try to use a different mathematical symbol for these objects.

Interlude VII - All these concepts at just one point - Really?

todo

Bundles and Fields

todo

The difference between Lie-, Covariant-, and Exterior-Derivative

TODO: rewrite paragraph from Wikipedia (https://en.wikipedia.org/wiki/Lie_derivative):

The Co-variant Derivative

A geometrically intuitive and very general explanation of the co-variant derivative can be simply given as: The co-variant derivative evaluates the directional derivative of a tensor in any direction at a point on a smooth manifold. This explanation is geometrically intuitive because humans usually have an easy time imagining a direction on a manifold and we also have an intuitive understanding of the directional derivative, as the thing that describes the rate of change of something. The above explanation is very general, firstly, because it specifies the ‘something’ that changes in a given direction as (nothing less than) a tensor. A tensor is a very general concept that includes scalars, vectors, co-vectors and products of them. Secondly, the generality of the co-variant derivative stems from its definition on (nothing less than) a smooth manifold. Again a smooth manifold is a very general concept that includes high dimensional curved spaces, but also more familiar special cases like the 3D Euclidean space. This generality of the co-variant derivative means that the mathematical definition is immensely valuable since it encompasses all kinds of special cases of directional derivatives. Simple cases like the directional derivative of a vector field in Euclidean space are equally well described as more complicated cases as directional derivatives of tensor fields on smooth manifolds in general. So instead of learning many different specializations of directional derivatives that are only valid for certain cases - we shall learn ‘the one’ directional derivative that rules them all.

The Lie-Derivative

Another kind of directional derivative that ‘lives’ on a smooth manifold is the Lie derivative. TODO

Relation between Co-variant Derivative and Lie Derivative

Note that the antisymmetrized covariant derivative ∇uv − ∇vu, and the Lie derivative Luv differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization is the Lie derivative.
TODO

Jacobian Sidenote:

By far the most accesible geometrical interpretation of the Jacobian matrix that I’ve come across is given in this video: https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3

If you need to brush up your linear algebra and want to develop some immensely helpful geometric intuition for matrices and vectors it pays off watching the whole series on linear algebra. In just over 2.5h you can binge learn the essence of geometrical understanding of linear algebra. Something that otherwise you could spend month trying to wrap your head around. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Nabla and all Kinds of Directional Derivatives

The Nabla operator is so important that it is hardly impossible to overstate its importance. It is the operator that generalizes directional derivatives of different kinds on manifolds. Directional derivatives are the objects that allow us to talk about rates of change and arguably this is a large portion of what physics (or if you want to get philosophical - the world as it is perceived by us) is about.

It is also interesting that Nabla isn’t just another symbol borrowed from the greek alphabet, but a symbol that was invented by […] TODO. “physical mathematics is very largely the mathematics of ∇”

Nabla is arguably a confusing object. It has many meanings and takes different forms and shapes and therefore was misinterpreted in many books and publications. There is even a report [Tai 1994] of the improper uses of Nabla in the literature that lists more than 60 books on math, physics and engineering that use the operator sloppily or wrongly. The list of publications that makes improper use of the operator must go in the hundreds if not thousands.

Other concepts and operators that are related to Nabla include the gradient, divergence, curl, Laplace, the derivative, the vector differential operator, the directional derivative, the covariant derivative, connections on a manifolds, parallel transport, etc.

There are also plenty of notations for some special cases of this operator which includes…

It is obvious that Nabla is a very special and important object with lots of uses, confusions, and specializations. Specializations come in the form of either restricting it to a certain coordinate system or to certain domains. These specializations are the source of many errors, inprecissions, and confusions. When a specialization of the Nabla operator is taken out of its context it might easily be wrongly interpreted or applied.

Here, we will try to be extra careful and define the nabla operator in an unambiguous, general sense, and enumerate the possible specializations of the operator.

Nabla as a Symbol

The nabla operator as a symbol has three slots, (\nabla_{\square}(\square))_{\square} The first slot is open for a vector (i.e., an element of the tangent vector space T_pM at a given point p), the second slot is for the operand which needs to be a tensor field on the manifold, and the third slot is open for a point on the manifold. By filling certain slots and leaving other slots open we can get specializations of the nabla operator in very different ways and with very different meaning. Before we do that we will try to get a high-level understanding of the meaning of the nabla operation, i.e., the nabla operator with all three slots filled.

The meaning of the operation (\nabla_{v}(s)){p} is that of a directional derivative of the operand s at p, where v gives the direction. It gives at a given point p a derivative of the field s. So the result of the operation is of the same kind as the field of the operand. If the operand s is a scalar field, the result of (\nabla{v}(s)){p} is a scalar; if s is a vector field, the result of (\nabla{v}(s)){p} is a vector; … The operand s must be a tensor field which of course also includes scalar fields, vector fields, and co-vector fields, on the manifold. The actual derivation, i.e., the rule how to compute (\nabla{v}(s))_{p} is not uniquely defined, but can be prescribed in different ways. This non-unique prescription is called the connection on the manifold. Once it is prescribed the connection tells us how to connect different tangent spaces with each other and we can think of it as “part of the manifold”. We will look at different prescriptions (i.e., connections) later and see that some have nicer properties than others. For now we want to look a bit closer at what the symbol nabla can mean without worrying about how to compute it.

Back to the three slots of the nabla operation. Interestingly, when specializing the nabla operator, we can choose to leave any of the three slots open or unassigned. If we leave the third slot open we turn the nabla operator into an operator that is defined for vector fields on the manifold, that is evaluated at every point for the local tangent vector space. However, instead of a vector in the first slot we now need a vector field in the first slot. This is very commonly done and conveniently allows us to specify operators that operate on whole vector, and tensor fields of the manifold.

If the operand is a scalar function f \in C(\infty(M)) on the manifold and the vector is an element of the tangent vector space T_pM, the nabla operator gives us the directional derivative of the function at point p.

If we leave the second (i.e., the operand) slot open we get the directional derivative operator \nabla_{v}(\square) that takes a function on the manifold and evaluates the directional derivative. Interestingly, we can also leave the first (i.e., the direction) slot open and write \nabla_{\square}(f), which gives us another operator. This operator is called the gradient of the function f. It takes a vector as input and gives us again the directional derivative of f at p.

Analogously we can define different specializations of the nabla operator by reserving the second (i.e., the operand) slot for a vector. The operation then results in a vector at point p or in a vector field if we apply it to the whole manifold. By leaving the first slot open we get an operator that takes a vector as input and gives the directional derivative of the vector field in the second slot. By leaving the second slot open it gives the directional derivative of one vector field in the direction of another.

TODO: what are the common names for these? what is div, curl?

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/7869/bad1475.0001.001.pdf?sequence=5&isAllowed=y

Directional Derivative of Tensor Fields

Schuller WEHIWSGL-7, 556

Embedding and Immersion

A smooth manifold can be embedded or immersed in \(\rm \mathbb R^d\), for some \(d\in \mathbb N\).

A smooth map \(\phi: M \to N\) is an immersion of \(M\) into \(N\) if the derivative is injective, for all \(p\in M\).

The manifold \(M\) is said to be an immersed submanifold of \(N\).

A smooth map \(\phi: M \to N\) is said to be a smooth embedding of \(M\) into \(N\) if

• \(\phi: M\to N\) is an immersion
• \(M \cong_{\mathrm{top}}\phi(M)\subseteq N\),

where \(\phi(M)\) carries the subset topology inherited from \(N\).

The manifold \(M\) is an embedded submanifold of \(N\).