Riemannian geometry I

I had many courses in my undergraduate studies, but for one reason or another never got to see some riemannian geometry. Which is really a shame, because I always wanted to learn some general relativity. So, here I am, studying riemann geometry for the first time. Since there’s lots of things in it, what follows is really just a summary, primarily for my own understanding.

Let {M} be a smooth manifold of dimension {n}, {TM = \sqcup_{p\in M}{T_p M}} his tangent bundle, {T^\ast M} the cotangent bundle, and let {\mathcal{T}(M)} denote the space of smooth sections of {TM} (i.e. the smooth vector fields on {M}). Analogously, {\mathcal{T}^{k}_{l}(M)} will denote the smooth sections of the tensor bundle {T^{k}_{l}M} – that is, the smooth tensor fields [1]. Tensors fields are multilinear with respect to {C^{\infty}(M)}. For example, {\mathcal{T}M = \mathcal{T}_{1}(M)}, while {\mathcal{T}^{1}(M)} are the {1}-forms. We also impose {\mathcal{T}^{0}(M)=C^{\infty}(M)}.

1. Riemannian metrics

A Riemannian metric on {M} is a {2}-tensor field (in {\mathcal{T}^2 (M)}) – given in local coordinates by {g_{ij}dx^i \otimes dx^j} – which is:

  1. symmetric: {g(X,Y) = g(Y,X)} for every {X, Y \in \mathcal{T}(M)};
  2. positive definite: {g(X,X)>0} for every {X\neq 0}.

It induces an inner product on every tangent space. It also allows to turn a vector field into a 1-form and viceversa, namely (in components)

\displaystyle (X^{\flat})_j:=g_{ij} X^i \quad\quad\mbox{ and }\quad\quad (\omega^{\sharp})^{j}:= g^{ij}\omega_i ,

with the obvious extensions to tensors [2]. Basically {X^{\flat}(\cdot) = g(X,\cdot)}. One can define and extend the trace operator as well: it’s the contraction with respect to a pair of covariant-contravariant indices. But {g} allows to define it for pairs of covariant indices as well: if {a \in\mathcal{T}^2 (M)} then one defines for example with respect to the first index

\displaystyle \mathop{\mathrm{Tr}}_g a := \mathop{\mathrm{tr}} a^\sharp = \mathop{\mathrm{tr}} g^{ij}h_{ik} = g^{ij}h_{ij},

and one can extend to operators of arbitrary rank (obviously the indices w.r.t. which one takes contraction have to be specified).

Of course one is interested in the general properties of riemannian manifolds, so there’s an obvious notion of equivalence given by the definition of isometry: a diffeomorphism {\psi: \, M \rightarrow \tilde{M}} is an isometry between spaces {(M,g)}, {(\tilde{M},\tilde{g})} if

\displaystyle \psi^\ast \tilde{g} = g.

If {M} is oriented, then the riemannian metric induces the riemannian volume element

\displaystyle dV = \sqrt{\det g}\; dx^{1}\wedge \ldots \wedge dx^{n},

unique up to scalar multiples. There’s nothing special about the coordinates, so if {\{E_i\}_{i=1}^{n}} is any local frame and {\{\varphi^{i}\}_{i=1}^{n}} the dual frame, one can use {g_{ij}=g(E_i,E_j)}, and the volume element is again

\displaystyle dV= \sqrt{\det g}\; \varphi^1 \wedge\ldots\wedge \varphi^n.

This choice has the effect that {dV(E_1, \ldots, E_n)=1} when the {E_i}‘s are an oriented orthonormal basis for {T_p M}.

On an orientable riemannian manifold {M} (with or without boundary) one can rebuild the usual machinery for the Green-Gauss formulas. Define {\mathop{\mathrm{grad}}f} by

\displaystyle df(X) := g(\mathop{\mathrm{grad}}f, X),

and divergence operator {\mathop{\mathrm{div}}X} by

\displaystyle \mathop{\mathrm{div}}X\; dV := d(i_X dV),

where {d} is the exterior derivative and {i_X} the interior multiplication by {X}, i.e. {i_X \omega (V_1,\ldots, V_k) = \omega (V_1,\ldots, V_k, X)}. Then thanks to Stoke’s theorem one can prove

\displaystyle \int_{M}{\mathop{\mathrm{div}}X}\,dV = \int_{\partial M}{g(X,N)}\,d\tilde{V},

where {d\tilde{V}} is the volume element arising from the induced metric on {\partial M}, and {N} is the outward normal to {\partial M}. It’s defined by {\mathop{\mathrm{Span}}(N(p)) = {T_p M}/{T_p \partial M}}. Then, since

\displaystyle \mathop{\mathrm{div}}(uX) = u \mathop{\mathrm{div}}X + g(\mathop{\mathrm{grad}}u, X)

one has

\displaystyle \int_{M}{u\mathop{\mathrm{div}}X}\,dV+\int_{M}{g(\mathop{\mathrm{grad}}u,X)}\,dV = \int_{\partial M}{u\, g(X,N)}\,d\tilde{V},

and defining the laplacian {\Delta u := \mathop{\mathrm{div}}(\mathop{\mathrm{grad}}u)}, Green formula follows

\displaystyle \int_{M}{u\Delta v}\,dV + \int_{M}{g(\mathop{\mathrm{grad}}u,\mathop{\mathrm{grad}}v)}\,dV = \int_{\partial M}{u\, Nv}\,d\tilde{V}.

With this one has the usual property that the eigenvalues of {\Delta} are non-positive. Harmonic functions on compact connected manifolds without boundary are constants, by Green formula and positive definiteness of {g}.

2. Connections

There’s another structure that can be produced on a manifold {M}, a connection. A (linear, or affine) connection is an operator

\displaystyle \nabla\,:\, \mathcal{T}(M)\times \mathcal{T}(M) \rightarrow \mathcal{T}(M)

such that

  1. it is {C^\infty (M)}-linear in the 1st variable;
  2. it’s {\mathbb{R}}-linear in the 2nd variable;
  3. the following product rule holds:

    \displaystyle \nabla_X (f Y) = f\nabla_X Y + Xf\, Y.

It’s easy to see that {(\nabla_X Y)(p)} only depends on the value {X(p)} and on the values of {Y} in an arbitrarily small neighbourhood of {p} (we see here the effect of the different linearity in the two variables). In any given frame {\{E_i\}}, one can compute the Christoffel symbols

\displaystyle \nabla_{E_i}{E_j} =: \Gamma_{ij}^{k} E_k,

and then one can express {\nabla_X Y} in coordinates ( {X = X^i \partial_i}, the Christoffel symbols calculated in the same coordinate frame) by

\displaystyle \nabla_X Y = (X Y^k + X^i Y^j \Gamma_{ij}^k)\partial_k.

The previous relation actually allows to define a connection for every set of smooth functions {\Gamma_{ij}^{k}}. Notice that in the euclidean case of {\mathbb{R}^n}, the Christoffel symbols are identically {0}.

Given a connection one can extend it uniquely to maps

\displaystyle \nabla\,:\, \mathcal{T}(M)\times \mathcal{T}^{k}_{l}(M) \rightarrow \mathcal{T}^{k}_{l}(M)

by imposing the following further properties:

  1. if {f \in C^{\infty}(M)}, then {\nabla_X f = Xf};
  2. Product rule holds for tensor products:

    \displaystyle \nabla_X (F\otimes G) = \nabla_X F \otimes G + F \otimes \nabla_X G;

  3. traces/contractions commute with {\nabla_X}:

    \displaystyle \nabla_X \mathop{\mathrm{Tr}} F = \mathop{\mathrm{Tr}} \nabla_X F.

With these properties, one then has that a product rule also holds for the natural pairing of forms and vectors,

\displaystyle \nabla_X (\omega, Y) = (\nabla_X \omega, Y) + (\omega, \nabla_X Y),

and for a general tensor field {F \in \mathcal{T}^k_l (M)} it is

\displaystyle (\nabla_X F)(V_1,\ldots, V_k ;\omega_1, \ldots, \omega_l) = X(F(V; \omega))- \sum_{j=1}^{l}F(V;\omega_1,\ldots,\nabla_X \omega_j,\ldots, \omega_l )

\displaystyle -\sum_{i=1}^{k}F(V_1,\ldots,\nabla_X V_i,\ldots, V_k ;\omega).

As an example, in coordinates,

\displaystyle \nabla_X \omega = (X^i \partial_i \omega_j - X^i \omega_k \Gamma_{ij}^k)dx^j.

From a linear connection one can define an operator {\nabla\,:\, \mathcal{T}^{k}_{l}(M) \rightarrow \mathcal{T}^{k+1}_{l}(M)} called total covariant derivative as follows:

\displaystyle (\nabla F)(V_1,\ldots, V_k; X; \omega_1,\ldots, \omega_l):= (\nabla_X F) (V_1,\ldots, V_k; \omega_1,\ldots, \omega_l).

For example, if {u \in C^\infty (M)} then {\nabla u = du}. {\nabla(\nabla u) = \nabla^2 u } is called the covariant hessian of {u}, and it holds {\nabla^2 u (X,Y)= Y(Xu) - (\nabla_Y X)u}. From a coordinate point of view, one can write (commonly accepted notation is a very unfortunate one) in the case of a vector field {Y}:

\displaystyle \nabla Y = Y^i_{\;\; ;\,j}\; \partial_i\otimes dx^{j} = (\partial_j Y^i + Y^k \Gamma_{jk}^i)\;\partial_i\otimes dx^{j}.

As it will be useful in what follows, here’s the coordinate expression for {\nabla g} (it holds for any 2-tensor actually):

\displaystyle g_{ij\,;\;k} = \partial_k g_{ij} - g_{lj}\Gamma_{ki}^l - g_{il}\Gamma_{kj}^l.

3. Geodesics

The importance of a connection is in that it allows one to define geodesics. One might be tempted to define them through the usual length minimization properties, but that’s a global feature, and we rather need a local one. Using a physical analogy, a point along a geodesic experiences no acceleration. But acceleration, unlike velocity, can’t be calculated by taking the limit of a quotient – as the numerator would contain the difference of vectors that live in different tangent spaces, which makes no sense. A connection allows to avoid this problem by defining derivations along curves.

Let {\gamma: I \rightarrow M} be a smooth curve in {M}, and let {V\in \mathcal{T}(M)} Then we can define the covariant derivative {D_t} along {\gamma} at {\gamma(t)} as

\displaystyle D_t V := (\nabla_{\dot{\gamma}(t)} V)(\gamma(t)).

The definition is well posed because actually {\nabla_X Y (p)} only depends on {X(p)} and the values of {Y} along a curve through {p} with tangent {X(p)}, rather than from an entire neighbourhood of {p} (it’s immediate from the coordinate expression for {\nabla_X Y}). If the vector field {V} is defined only on {\gamma} that’s no trouble, since it can always be extended to a vector field on {M}. This definition implies the product rule

\displaystyle D_t (fV) = \dot{f}V + f D_t V,

where {\dot{f}(t) = (f\circ \gamma)^\prime (t)}. In local coordinates

\displaystyle D_t V = (\dot{\gamma}^i \partial_i V^k(\gamma) + \dot{\gamma}^iV^j(\gamma)\Gamma_{ij}^k (\gamma))|_t \;\partial_k = (\dot{V}^k + \dot{\gamma}^iV^j\Gamma_{ij}^k (\gamma))|_t \;\partial_k.

Now, acceleration is {D_t \dot{\gamma}}, thus the condition for {\gamma} to be a geodesic is that

\displaystyle D_t \dot{\gamma} = 0 \quad\quad\quad \forall t \in I,

which can be immediately translated into coordinates:

\displaystyle \ddot{\gamma}^k(t) + \dot{\gamma}^i (t)\,\dot{\gamma}^j (t)\,\Gamma_{ij}^k (\gamma(t))=0.

These are the geodesic equations, which are ODEs. By standard ODE theory then, given initial conditions {\gamma(t_0)=p}, {\dot{\gamma}(t_0)=V\in T_p M}, one has that a geodesic through {p} with tangent {V} exists, and is unique in its maximal domain of existence.

In general, a vector field in {\mathcal{T}(\gamma)} such that {D_t V=0} is said to be parallel along {\gamma} – so, a geodesic is a curve whose velocity is parallel along the curve itself. The following holds: if {V_0 \in T_p M}, and {\gamma} is a curve, there exists a unique vector field in {\mathcal{T}(\gamma)} parallel along {\gamma} to {V_0} [3]. Thus given a curve {\gamma} one can define the operator of parallel transport from {\gamma(t_0)} to {\gamma(t_1)}

\displaystyle P_{t_0\,t_1}\,:\, T_{\gamma(t_0)}M \rightarrow T_{\gamma(t_1)}M,

which is a linear isomorphism of tangent spaces. With this operator one can actually compute {D_t} as the limit of a quotient:

\displaystyle D_{t_0} V = \lim_{t\rightarrow t_0} \frac{P_{t_0\,t}^{-1}V(t) - V(t_0)}{t-t_0}.

4. Riemannian connection

Given a linear connection {\nabla} one can define the following tensor field in {\mathcal{T}^2 (M)},

\displaystyle \tau (X,Y) = \nabla_X Y - \nabla_Y X - [X,Y],

called the torsion tensor. It is indeed a tensor field as it’s {C^\infty(M)}-linear in both variables. By writing down local coordinates, one sees that {\tau} is identically 0 iff the Christoffel symbols are symmetric in the lower indices (in a coordinate frame, not necessarily in any frame [4]), in which case we say {\nabla} is symmetric. All in all, the torsion tensor measures how much $\nabla$ fails to be symmetric.

So far {g} and {\nabla} are separate objects, and as we’ve seen there’s plenty of choices for {\nabla} itself. We boil it down to just one by imposing two conditions:

  1. {\nabla} is compatible with {g}, i.e.

    \displaystyle \nabla_X (g(Y,Z)) = g(\nabla_X Y,Z) + g(Y, \nabla_X Z);

  2. {\nabla} is symmetric.

The 1st condition is equivalent to a number of other conditions:

  1. {\nabla g \equiv 0};
  2. if {V,W \in \mathcal{T}(\gamma)} for some curve {\gamma} then

    \displaystyle \frac{d}{dt} g(V,W) = g(D_t V, W) + g(V, D_t W);

  3. if {V,W} are parallel along {\gamma} then {g(V,W)} is constant along {\gamma};
  4. parallel transport operators {P_{t_0\;t_1}} are all isometries.

For example, the 1st one is equivalent because {(\nabla g)(X,Y,Z) = X(g(Y,Z)) - g(\nabla_X Y, Z) - g(Y,\nabla_X Z)}. As for the 2nd, extending {V,W} to vector fields {\tilde{V}, \tilde{W}} and taking {X=\dot{\gamma}(t)} the compatibility relation implies

\displaystyle \dot{\gamma}(t) g(\tilde{V},\tilde{W}) = g(\nabla_{\dot{\gamma}(t)}\tilde{V},\tilde{W}) + g(\tilde{V},\nabla_{\dot{\gamma}(t)}\tilde{W}),

which is exactly the condition in 2., since {\dot{\gamma}(t) g(\tilde{V},\tilde{W}) = \frac{d}{dt} g(V,W)}. The other way round is similar. Now {2. \Rightarrow 3. \Leftrightarrow 4.} is immediate, and finally {3.\Rightarrow 2.} by taking the derivative of {g(P_{t_0\,t}V_0,P_{t_0\,t}W_0)= g(P_{t_0\,t}V_0,P_{t_0\,t}W_0)+g(V(t),W(t))-g(P_{t_0\,t}V_0,W(t))-g(V(t),P_{t_0\,t}W_0)}.

As stated above, compatibility + symmetry completely define a unique connection, the riemannian connection (or Levi-Civita’s):

Theorem 1 (Fundamental lemma of Riemannian geometry) Let {(M,g)} be a riemannian manifold. Then there exists a unique connection {\nabla} that is compatible with {g} and symmetric.

Proof: By compatibility

\displaystyle X g(Y,Z) = g(\nabla_X Y,Z) + g(Y, \nabla_X Z)

and cyclic permutations, but by symmetry

\displaystyle X g(Y,Z) = g(\nabla_X Y,Z) + g(Y, \nabla_Z X) + g(Y,[X,Z]).

Thus adding a cyclic permutation and subtracting the other

\displaystyle g(\nabla_X Y, Z) = \frac{1}{2}(X g(Y,Z) + Y g(Z,X) - Z g(X,Y)

\displaystyle + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])),

where as you can notice the r.h.s. doesn’t depend on {\nabla} at all (which implies uniqueness). Now, choosing {X=\partial_i, Y=\partial_j, Z=\partial_l},

\displaystyle g(\nabla_{\partial_i} \partial_j, \partial_l)=g_{kl}\Gamma_{ij}^k = \frac{1}{2}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}),

because {[\partial_i,\partial_j] =0}, thus

\displaystyle \Gamma_{ij}^k = \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}).

Symmetry is evident, compatibility may be directly verified by substituting into the previous expression for {g_{ij\,;\,k}}. \Box

As a corollary, by the equivalent condition stated above, geodesics w.r.t. the riemannian connection have constant speed:

\displaystyle \frac{d}{dt} g(\dot{\gamma},\dot{\gamma}) = 0.

In the related field of pseudo-riemannian geometry, where the positive definiteness of {g} is relaxed to non-degeneracy, virtually all results that don’t require {g} to be positive hold as well. In applications to physics (general relativity) one considers 4-dimensional manifolds with metrics of signature {(-1,1,1,1)}. The pseudo-riemannian connection is built the same way, and thus geodesics exhibit the same constant speed property, but now it can be that {g(\dot{\gamma},\dot{\gamma}) >, <, = 0}. In the case {g(\dot{\gamma},\dot{\gamma})=0} one calls the geodesic null-like: this is interpreted as the path a ray of light would take. Speed of light is thus constant when measured by any observer.

Notice also that at this point we are already able to compute (at least numerically, hopefully) the geodesics of a manifold, given {g}: first compute the Christoffel symbols of the Levi-Civita’s connection, then solve the geodesic equations for given initial values.

{[1]} {T^{k}_{l}M= \sqcup_{p\in M}{T^{k}_{l}(T_p M)}}, where {T^{k}_{l}(V) = {V^{\ast}}^{\otimes k}\otimes {V^{\otimes l}}}. Just to clarify, {A\in T^{k}_{l}(V)} is a real-valued multilinear map which takes as arguments {k} vectors and {l} covectors.

{[2]} It is customary to denote by {g^{ij}} the inverse matrix of {g_{ij}}. Its elements are the coefficients of a {\mathcal{T}_2 (M)} tensor, {g^{ij} \partial_i\otimes\partial_j}.

{[3]} The result again is a consequence of standard ODE theory, but notice that one has existence along all of the curve because the system is linear in {V}.

{[4]} One can work out the formula for the Christoffel symbols’ transformation law under a change of coordinates {x^i(p) \rightarrow y^j(p)}, namely

\displaystyle \tilde{\Gamma}_{ij}^{k} = \left(\frac{\partial x_m}{\partial y_i}\frac{\partial x_l}{\partial y_j}\right)\frac{\partial y_k}{\partial x_s}\;\Gamma_{ml}^s + \frac{\partial^2 x_s}{\partial y_i \partial y_j}\frac{\partial y_k}{\partial x_s}.

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