This section based on. For convenience, we always assume the manifold is path-connected, since if it is not, then we can study each component separately.
Flat affine geometry \mathbb A^n is \R^n regarded as an affine geometry. It is flat, that is, with curvature zero. There is only one
k-simplex, one
k-ellipsoid, and one
k-parallelopiped for each of k = 1, 2, \dots, n. There is no length of a line segment, but the length ratio of two
directed line segments on the same line is well-defined. Similarly, the separation between two parallel planes is not well-defined, but the ratio of directed separations is well-defined. The centroid of a compact subset is well-defined.
Geodesics An affine manifold has an
affine connection (by default assumed to be
torsion-free), and can be written as (M, \nabla). It allows defining many structures that appears also in Riemannian geometry. First,
covariant derivative or
parallel transport is defined by \nabla, and is essentially equivalent to the connection itself. This then allows
geodesics to be defined as curves \gamma such that \nabla_{\dot \gamma} \dot\gamma is parallel to \dot \gamma. Every geodesic has an affine parameterization that is unique up to an affine change of variables. The
affine parameterization satisfies \nabla_{\dot \gamma} \dot\gamma = 0, or more succinctly, \ddot \gamma = 0 . Using the affine parameterization, the ratio of two directed segments on a geodesic is well-defined. Geodesics can thus be thought of as straight embeddings of the affine line \mathbb A^1. Note that it is meaningless in general to speak of a constant-speed curve when the curve is not a geodesic, because we cannot compare the length-ratio of two vectors not lying on the same affine line. A geodesic is
complete iff it is extended forwards and backwards for all time. That is, it has an affine parameterization of type \R \to M. If all its geodesics are complete, then (M, \nabla) is
geodesically complete. If there is no way to complete it, i.e. no geodesically complete affine manifold \bar M such that M \subset \bar M, then M is
essentially geodesically incomplete. At any point x \in M and any direction v \in T_xM there exists a unique affinely parameterized geodesic \gamma such that \gamma_v(0) = x, \; \dot\gamma_v(0) = v. Thus, if we arbitrarily fix a constant t_0 > 0, then the map v \mapsto \gamma_v(t_0) defines the
exponential map, which maps an open neighborhood of zero in T_x M to M itself. The precise choice of t_0 does not matter, but there is no natural choice in general, since unlike in Riemannian manifold case, there is no natural way to measure distance on a tangent space. For convenience, we arbitrarily fix t_0 = 1, and define the exponential map as \exp_x(v) = \gamma_v(1).
Killing An
affine Killing vector field is a vector field X that preserves the affine connection. That is, \mathcal L_X(\nabla) = 0, where \mathcal L is the
Lie derivative. In particular, infinitesimal parallelopipeds are still parallelopipeds when flowing under the Killing vector field. The set of affine Killing vector fields on (M, \nabla) with the
Lie bracket [\cdot, \cdot] makes for a
Lie algebra \mathfrak{K}(M, \nabla). A vector field is
complete iff the integral curves exist for all time. A complete Killing vector field generates a 1-parameter family of affine diffeomorphisms of M to itself. (M, \nabla) is
affine Killing complete iff every affine Killing vector field is complete. The group of affine diffeomorphisms mapping (M, \nabla) to itself is \operatorname{Aff}(M, \nabla). It is a finite-dimensional
Lie group. Its
Lie algebra \mathfrak{aff}(M, \nabla) is the Lie subalgebra of \mathfrak{K}(M, \nabla) consisting of the affine Killing complete vector fields.
Morphisms An
affine diffeomorphism is a
diffeomorphism that preserves the affine connection. A
local affine diffeomorphism is an affine diffeomorphism restricted on only an open subset. An affine
covering map is a function between two affine manifolds f: M \to N, such that any x \in M has an open neighborhood in which f restricts to an affine diffeomorphism. An affine manifold is (locally)
symmetric iff for any two points x, y \in M, there exists a (local) affine diffeomorphism mapping x to y. For example, any (locally) symmetric pseudo-Riemannian space is automatically (locally) symmetric as an affine manifold. Intuitively, a locally symmetric manifold is one that, if one stands at any point of it, and looks out from it, one cannot discover where one is (unless one looks so far that global geometry comes to play). Similarly, a symmetric manifold is one that, if one stands at any point of it, and looks out from it, one cannot discover where one is, no matter how far one looks. Given a smooth submanifold N \subset M, it is
totally geodesic iff any x \in N, \; v \in T_x N, the geodesic in M passing (x, v) is entirely contained in N. For such manifolds, the affine connection on the bigger manifold
is an affine connection on the submanifold, so the submanifold is an affine manifold in a natural way. Explicitly, given a point x \in N, and a direction to transport v \in T_x N, we transport T_xN as if it is a subspace of T_x M transported along v \in T_x M using a geodesic curve \gamma(t). The transported T_xN becomes a subspace of T_{\gamma(t)}M. Because N is totally geodesic in M, the transported T_xN is equal to T_{\gamma(t)} N.
Immersion In general, a differentiable manifold immersion f : N \to M is not totally geodesic, in that the geodesics of M that start on f(N) may immediately leave f(N). In the Riemannian case, where a
second fundamental form bends geodesics in M to the nearest curves in f(N). In the affine case, we can similarly define
affine fundamental forms, though there is no longer a unique natural choice. Suppose that (N, \nabla), (M, \nabla') are affine manifolds. At any x \in N, we pick some V_x such that T_{f(x)} M = f^*(T_xN) \oplus V_x, and the choice of V_x varies smoothly as x \in N varies. The subspaces V_x can be interpreted as a choice of
transverse directions to the immersion. There is no natural choice unlike in Riemannian geometry, since perpendicularity is undefined. An
affine fundamental form is some multilinear map \alpha that takes two vector fields X, Y on N and produces a vector field in V, such that\nabla'_X f_*(Y) = f_*(\nabla_X Y) + \alpha(X, Y)In words, it states that the affine transport on (N, \nabla) can be equivalently performed by doing the affine transporting in (M, \nabla'), plus a correction in a transverse direction. We say f: N \to M is an
affine immersion iff there exists such an affine fundamental form. An affine immersion is totally geodesic iff its affine fundamental form is zero.
Coordinates Given a local
coordinate chart (x^1, \dots, x^n) : M \supset U \to \R^n, it produces a set of vector fields \partial_1, \dots, \partial_n spanning the tangent bundle. The
Christoffel symbols are defined in the same way as in Riemannian geometry:\nabla_{\partial_i} \partial_j = \Gamma_{ij}^k \partial_kIn Riemannian geometry, the
normal coordinates at a point p \in M causes the Christoffel symbols of the
Levi-Civita connection to vanish at that specific point. It is constructed using the geodesics via the exponential map \exp_p : T_p M \supset U \to M. This construction still works for an affine connection. In this way, one can regard M as "locally equivalent to \mathbb A^n", since \mathbb A^n \cong T_pM. This construction requires torsion-freeness, because normal coordinates exist iff the torsion is zero. This is one reason affine geometry usually assumes that the affine connection is torsion-free. Instead of using coordinates, one can also use the
vielbein formalism.
Curvature In affine geometry, as in Riemannian geometry, curvature encodes all
local features of affine geometry. Specifically, there is a (1, 3)-tensor called the
curvature tensor, defined in the same way:R(X, Y) := \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}Similarly, there is a (0, 2)-tensor obtained by
contracting the tensor, called the
Ricci curvature tensor:\operatorname{Ric}(X, Y) := \operatorname{Tr}(Z \mapsto R(Z, X)Y)though unlike in Riemannian geometry, the Ricci curvature tensor could be
asymmetric. Consequently, it splits to a
symmetric and antisymmetric part. An affine manifold is
flat iff its curvature tensor is zero. Similarly for
Ricci-flat. \operatorname{Ric} is symmetric on an open subset U \subset M if and only if there exists a
volume form on U invariant under parallel transport. If there is an affinely parameterized geodesic \gamma : (t_0, t_1) \to M such that \operatorname{Ric}(\dot \gamma, \dot \gamma) \to \pm\infty at either end, then there is no way to extend it further along that end, and thus it is
essentially incomplete (rather than
incidentally). Statements: • If \nabla R = 0, then (M, \nabla) is locally symmetric. • If \operatorname{Ric} is full-ranked, then \operatorname{Ric} can be interpreted as a
pseudo-Riemannian form, making (M, \operatorname{Ric}) a pseudo-Riemannian manifold, though its Levi-Civita connection might not be \nabla. • If \nabla R = 0 and \operatorname{Ric} is full-ranked, then the Levi-Civita connection of (M, \operatorname{Ric}) is \nabla. Thus, in this case, the affine manifold is canonically
just a locally symmetric pseudo-Riemannian manifold. • (M, \nabla) is locally symmetric if and only if at each point x the
geodesic symmetry y \mapsto \exp_x(-\exp_x^{-1}(y)) is an affine diffeomorphism. Note that the geodesic symmetry generalizes
point reflection. More generally, An affine manifold (M, \nabla) is
metrizable iff we can impose a pseudo-Riemannian form g on M, such that \nabla is the Levi-Civita connection for g. Note a subtlety with the Blaschke metric. For any smooth orientable hypersurface in M \subset \R^n, we can define the equiaffine arclengths of curves on M using the Blaschke metric, which then makes M into a Riemannian manifold. But the Levi-Civita connection for the Blaschke metric may not be the Blaschke connection. Indeed, if the surface is strictly convex but not a quadratic surface, then they are not the same connection.
Affine surfaces If M has 2 dimensions, it is an affine surface. Affine surfaces have special properties. If \nabla R = 0, then \operatorname{Ric} is symmetric. Unlike the statement in the previous section, this does not require the assumption that \operatorname{Ric} is full-ranked, so (M, \operatorname{Ric}) might not be a pseudo-Riemannian manifold. As for Riemannian surfaces, an affine surface is flat iff it is Ricci-flat. == Problems ==