What follows are the definitions you need to go straight from real analysis to differential geometry.
Let be a set points, and a set of subset of . is a topological space if
(a) It contains and the empty set
(b) It is closed under finite intersections
(c) It is closed under arbitrary unions
A homeomorphism is a continuous bijection between topological spaces that has a continuous inverse. Let and be topological spaces, be a homeomorphism, and let and elements of the respective topological spaces. Then since is bijective. Also,
Similar reasoning gives
is a bijection, and the topological operations, intersections and unions, act the same way in and . This means that preserves the topological properties of .
Exercise: Convince yourself that the interior of the -dimensional ball is homeomorphic to , but the -dimensional ball itself is not.
A chart is a homeomorphism from an open subset of to . The chart is often written as the pair .
An atlas is a collection of charts such that .
Topological manifolds of dimension are topological spaces that additionally satisfy the following properties:
(a) M is a Hausdorff Space: Every pair of points and in can be separated into disjoint neighborhoods.
(b) M is second-countable: There is a countable basis for M’s topology
(c) Locally, M “looks” like . More specifically, every point has a neighborhood U that is homeomorphic to
We will need some additional structure to talk about differentiation on manifolds, because homeomorphisms do not preserve differential properties. For example, the function is a homeomorphism, and is differentiable, but is not differentiable at the origin. Differentiability is easy to define for euclidean space, so we will try to extend that definition to arbitrary manifolds.
Let and be two charts of a manifold. Consider the homeomorphism . Its domain and range are both , so it makes sense to talk about whether it is smooth. We say that the two charts are smoothly compatible if either or is smooth. Finally, we define a smooth manifold to be a manifold which has an atlas such that any two charts are smoothly compatible.
A smooth function has the property that for every chart on , is smooth.
A smooth map has the property that for every pair of charts of and of , is smooth.
It can be proven that functions and maps need only satisfy their composition properties for a particular choice of smooth atlas.
A smooth curve in is a smooth map where is an interval in .
If is a smooth curve, we define the tangent vector to at to be .
A diffeomorphism is a smooth bijection between two smooth manifolds that also has a smooth inverse. You may notice a similarity in the definitions of diffeomorphisms and homeomorphisms – just interchange “smooth manifold” with “topological space” and “smooth” with “continuous”.
Note that the maps from a smooth atlas of a smooth manifold are diffeomorphisms, since both and must be smooth.
If we want to apply our definition of a manifold to spaces with boundaries, we run into a slight issue. For example, if you take a point on the boundary of a closed sphere, the neighborhoods of that point do not look like – they look like one half of , such as the set . So we generalize our definition of smooth manifolds to that of smooth manifolds-with-boundary, defined such that all interior points must still have a neighborhood diffeomorphic to , but points on the boundary must have a neighborhood diffeomorphic to .
Let’s define the tangent space at a point of manifold . A derivation at is a linear map such that . is the vector space of all derivations at , and we call members of tangent vectors. Two tangent vectors and are orthogonal if . See here for a more thorough introduction to the tangent space.
The Inverse Function Theorem states that if is a smooth map between manifolds such that is a linear isomorphism at , then there exists an open neighborhood of on which is a diffeomorphism. Intuitively, if is an isomorphism at a point, it must be an isomorphism over a neighborhood, and linear maps with full rank are included by diffeomorphisms.
The Inversion Functon Theorem is actually a special case of an even more useful theorem, the Constant Rank Theorem. Suppose is a smooth map with constant rank . Then there exists a choice of local coordinates on and on such that
The tangent bundle is the set of all tangent vectors in a manifold.
Let M and N be differentiable manifolds and a differentiable map between them. Then is a submersion if at every point , is a surjective map from to . This definition implies , so we can imagine intuitively that a submersion looks locally like a projection from a higher-dimensional vector space to a lower dimensional vector space.
If is injective for all , then is called an immersion. Equivalently, is an immersion if . is called proper if for every compact subset of , is compact. An embedding is a proper, injective immersion. An alternative definition is that an embedding is an injective immersion that is also a homeomorphism.
Example: If is a smooth curve, is an immersion if and only if for all .
Proof: Note that is not full rank at if and only if there is some nonzero such that . And since , such an exists if and only if for some .
A vector field is a smooth map on a manifold such that .
A Riemannian Metric on a smooth manifold is a smooth, positive-definite, symmetric 2-form acting on the tangent space. A manifold paired with its metric is often denoted . The smoothness condition here means that if and are smooth, locally defined vector fields, then the map must be smooth. If one drops the positive-definite condition, one gets a Pseudo-Riemannian Metric on the manifold. The Lorentz Metric is pseudo-Riemannian, while the Euclidean Metric is Riemannian.
An isometry from to is a diffeomorphism such that for any two vector fields , . Just as homeomorphisms preserve topological structure and diffeomorphisms preserve smooth structure, isometries preserve geometric structure.
A local parametrization of a submanifold of manifold is a smooth embedding for some whose image is an open subset of .
A vector bundle of rank over a manifold is a smooth manifold paired with a smooth, surjective map such that for each ,
(a) is a vector space. is called the fiber of over .
(b) There is a neighborhood of and a diffeomorphism such that , where is a projection onto the first factor. Also, restricted to must be an isomorphism that takes it to .
This precise definition from John Lee’s Smooth Manifolds is a bit convoluted. Intuitively, we are assigning a vector space to each , and we can choose a neighborhood of such that the vector spaces inside aren’t too different, since there is a diffeomorphism that takes them to . The cylinder is a so-called trivial bundle, where each point in is assigned the same vector space. The mobius strip is a nontrivial bundle, since the vector space at each point of eventually flips.
Given a manifold , a vector field on and a point , a flow is a curve defined by and for taking values in some open .
The tensor bundle of type on is defined as . Every tensor bundle is a vector bundle, since we can simply project each tensor onto its point in the manifold. A tensor field is a smooth section of a tensor bundle.
Given a smooth function , we define its exterior derivative by , where is a vector field.
If and are smooth manifolds and is a smooth map, the push-forward associated with is defined such that , where is a vector field.
If is a smooth map, then the rank of is the rank of
Given a push-forward , we can also define its dual , called the pullback. Pullbacks allow us to differentiate differential forms, as for any n-form , we can define .
A local frame for a manifold and vector bundle over is an ordered tuple of smooth sections of defined on some open subset of such that for each , the are linearly independent and span the fiber .
A topological group is a topological space whose points have a group structure. The unit circle is an example of a topological group, where the group operation simply adds the angles of two points.
An -dimensional manifold is called parallelizable if there exist vector fields globally defined on that form a basis for the tangent space at any given . Obviously, all one-dimensional manifolds are parallelizable. In fact, all topological groups are parallelizable, but the converse is not true – the 7-dimensional sphere is parallelizable, but is not a topological group (In fact, and are the only spheres which are topological groups – see here for a full proof). The hairy-ball theorem, a result from algebraic topology, says that no even-dimensional spheres have continuous tangent vector fields that are non-vanishing. The usual proof of the hairy-ball theorem requires the euler characteristic, a tool from algebraic topology.
Exercise: Use the hairy-ball theorem to convince yourself that even-dimensional spheres are not parallelizable, and thus are not topological groups.
Exercise: Show that the product of parallelizable manifolds is parallelizable, proving that .
In fact, is the -torus, and , where denotes the wedge sum of topological spaces. We can see how this makes sense using cell-decompositions, described here. Essentially, is a one-cell and a zero-cell, so is a two-cell, two one-cells, and a zero-cell. And is just two one-cells, so taking the quotient leaves a two-cell and a zero-cell, which is topologically .
Let be a k-form. is closed if and is exact if there is some form such that . There is a close relationship between a geometric property of a space, whether it has “holes”, and an algebraic property, whether it can contain forms which are closed but not exact. This is formalized by the deRham Cohomology. The k-dimensional deRham cohomology of a space is the quotient of the space of closed k-forms by the space of exact k-forms. For a great exposition on deRham Cohomology, see here.
Exercise: Prove that all exact forms are closed – in other words .
Let be the space of vector fields on . A connection is a map , denoted , such that is a tensor in and a derivation in . In other words, for any ,
For more about connections, see here
The Lie Bracket of two smooth vector fields and is defined such that for any , .