Editing Riemannian manifold (section)

== Connections, geodesics, and curvature ==

=== Connections ===
{{Main|Affine connection}}

An [[affine connection |(affine) connection]] is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

Let <math>\mathfrak X(M)</math> denote the space of [[vector fields]] on <math>M</math>. An ''(affine) connection''
: <math>\nabla : \mathfrak X(M) \times \mathfrak X(M) \to \mathfrak X(M)</math>
on <math>M</math> is a bilinear map
<math>(X,Y) \mapsto \nabla_X Y</math> such that
# For every function <math>f \in C^\infty(M)</math>, <math>\nabla_{f_1 X_1 + f_2 X_2} Y = f_1 \,\nabla_{X_1} Y + f_2 \, \nabla_{X_2} Y, </math>
# The product rule <math>\nabla_X fY=X(f)Y+ f\,\nabla_X Y</math> holds.{{sfn|Lee|2018|pp=89–91}}
The expression <math>\nabla_X Y</math> is called the ''covariant derivative of <math>Y</math> with respect to <math>X</math>''.

=== Levi-Civita connection ===
{{Main|Levi-Civita connection}}

Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the [[Levi-Civita connection]].

A connection <math>\nabla</math> is said to ''preserve the metric'' if
: <math>X\bigl(g(Y,Z)\bigr) =  g(\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
A connection <math>\nabla</math> is ''torsion-free'' if
: <math>\nabla_X Y - \nabla_Y X = [X,Y], </math>
where <math>[\cdot,\cdot]</math> is the [[Lie bracket of vector fields|Lie bracket]].

A ''Levi-Civita connection'' is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.{{sfn|Lee|2018|pp=122–123}} Note that the definition of preserving the metric uses the regularity of <math>g</math>.

=== Covariant derivative along a curve ===

If <math>\gamma : [0,1] \to M</math> is a smooth curve, a ''smooth vector field along <math>\gamma</math>'' is a smooth map <math>X : [0,1] \to TM</math> such that <math>X(t) \in T_{\gamma(t)}M</math> for all <math>t \in [0,1]</math>. The set <math>\mathfrak X(\gamma)</math> of smooth vector fields along <math>\gamma</math> is a vector space under pointwise vector addition and scalar multiplication.{{sfn|Lee|2018|p=100}} One can also pointwise multiply a smooth vector field along <math>\gamma</math> by a smooth function <math>f : [0,1] \to \mathbb R</math>:
: <math>(fX)(t) = f(t)X(t)</math> for <math>X \in \mathfrak X(\gamma).</math>

Let <math>X</math> be a smooth vector field along <math>\gamma</math>. If <math>\tilde X</math> is a smooth vector field on a neighborhood of the image of <math>\gamma</math> such that <math>X(t) = \tilde X_{\gamma(t)}</math>, then <math>\tilde X</math> is called an ''extension of <math>X</math>''.

Given a fixed connection <math>\nabla</math> on <math>M</math> and a smooth curve <math>\gamma : [0,1] \to M</math>, there is a unique operator <math>D_t : \mathfrak X(\gamma) \to \mathfrak X(\gamma)</math>, called the ''covariant derivative along <math>\gamma</math>'', such that:{{sfn|Lee|2018|pp=101–102}}
# <math>D_t(aX+bY) = a\,D_tX + b\,D_tY,</math>
# <math>D_t(fX) = f'X + f\,D_tX,</math>
# If <math>\tilde X</math> is an extension of <math>X</math>, then <math>D_tX(t) = \nabla_{\gamma'(t)} \tilde X</math>.

=== Geodesics ===
{{Main|Geodesic}}
{{multiple image | image1=Plane geodesic.svg  | image2=Sphere geodesic.svg | footer=In Euclidean space <math>\mathbb R^n</math> (left), the maximal geodesics are straight lines. In the round sphere <math>S^n</math> (right), the maximal geodesics are [[great circles]].}}

[[Geodesics]] are curves with no intrinsic acceleration. Equivalently, geodesics are curves that locally take the shortest path between two points. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds. An ant living in a Riemannian manifold walking straight ahead without making any effort to accelerate or turn would trace out a geodesic.

Fix a connection <math>\nabla</math> on <math>M</math>. Let <math>\gamma : [0,1] \to M</math> be a smooth curve. The ''acceleration of <math>\gamma</math>'' is the vector field <math>D_t\gamma'</math> along <math>\gamma</math>. If <math>D_t\gamma' = 0</math> for all <math>t</math>, <math>\gamma</math> is called a ''geodesic''.{{sfn|Lee|2018|p=103}}

For every <math>p \in M</math> and <math>v \in T_pM</math>, there exists a geodesic <math>\gamma : I \to M</math> defined on some open interval <math>I</math> containing 0 such that <math>\gamma(0) = p</math> and <math>\gamma'(0) = v</math>. Any two such geodesics agree on their common domain.{{sfn|Lee|2018|pp=103–104}} Taking the union over all open intervals <math>I</math> containing 0 on which a geodesic satisfying <math>\gamma(0) = p</math> and <math>\gamma'(0) = v</math> exists, one obtains a geodesic called a ''maximal geodesic'' of which every geodesic satisfying <math>\gamma(0) = p</math> and <math>\gamma'(0) = v</math> is a restriction.{{sfn|Lee|2018|p=105}}

Every curve <math>\gamma : [0,1] \to M</math> that has the shortest length of any admissible curve with the same endpoints as <math>\gamma</math> is a geodesic (in a unit-speed reparameterization).{{sfn|Lee|2018|p=156}}

==== Examples ====

* The nonconstant maximal geodesics of the Euclidean plane <math>\mathbb R^2</math> are exactly the straight lines.{{sfn|Lee|2018|p=105}} This agrees with the fact from Euclidean geometry that the shortest path between two points is a straight line segment.
* The nonconstant maximal geodesics of <math>S^2</math> with the round metric are exactly the [[great circles]].{{sfn|Lee|2018|p=137}} Since the Earth is approximately a sphere, this means that the shortest path a plane can fly between two locations on Earth is a segment of a great circle.

=== Hopf–Rinow theorem ===
{{Main|Hopf–Rinow theorem}}

[[File:Punctured plane is not geodesically complete.svg|thumb|The punctured plane <math>\mathbb R^2 \backslash \{(0,0)\}</math> is not geodesically complete because the maximal geodesic with initial conditions <math>p = (1,1)</math>, <math>v = (1,1)</math> does not have domain <math>\mathbb R</math>.]]

The Riemannian manifold <math>M</math> with its Levi-Civita connection is ''[[geodesically complete]]'' if the domain of every maximal geodesic is <math>(-\infty,\infty)</math>.{{sfn|Lee|2018|p=131}} The plane <math>\mathbb R^2</math> is geodesically complete. On the other hand, the [[punctured plane]] <math>\mathbb{R}^2\smallsetminus\{(0,0)\}</math> with the restriction of the Riemannian metric from <math>\mathbb R^2</math> is not geodesically complete as the maximal geodesic with initial conditions <math>p = (1,1)</math>, <math>v = (1,1)</math> does not have domain <math>\mathbb R</math>.

The [[Hopf–Rinow theorem]] characterizes geodesically complete manifolds.

'''Theorem:''' Let <math>(M,g)</math> be a connected Riemannian manifold. The following are equivalent:{{sfn|do Carmo|1992|pp=146–147}}
* The metric space <math>(M,d_g)</math> is [[Complete metric space|complete]] (every <math>d_g</math>-[[Cauchy sequence]] converges),
* All closed and bounded subsets of <math>M</math> are compact,
* <math>M</math> is geodesically complete.

=== Parallel transport ===
{{Main|Parallel transport}}
[[File:Parallel transport sphere2.svg|thumb|right|Parallel transport of a tangent vector along a curve in the sphere.]]

In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. [[Parallel transport]] is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.{{sfn|Lee|2018|pp=105–110}}

Specifically, call a smooth vector field <math>V</math> along a smooth curve <math>\gamma</math> ''parallel along <math>\gamma</math>'' if <math>D_t V = 0</math> identically.{{sfn|Lee|2018|p=105}} Fix a curve <math>\gamma : [0,1] \to M</math> with <math>\gamma(0) = p</math> and <math>\gamma(1) = q</math>. to parallel transport a vector <math>v \in T_pM</math> to a vector in <math>T_qM</math> along <math>\gamma</math>, first extend <math>v</math> to a vector field parallel along <math>\gamma</math>, and then take the value of this vector field at <math>q</math>.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the [[punctured plane]] <math>\mathbb R^2 \smallsetminus \{0,0\}</math>. The curve the parallel transport is done along is the unit circle. In [[Polar coordinate system|polar coordinates]], the metric on the left is the standard Euclidean metric <math>dx^2 + dy^2 = dr^2 + r^2 \, d\theta^2</math>, while the metric on the right is <math>dr^2 + d\theta^2</math>. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.

{{multiple image
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| header = Parallel transports on the punctured plane under Levi-Civita connections
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|image1=Cartesian_transport.gif
|width1=200
|caption1=This transport is given by the metric <math>dr^2 + r^2 d\theta^2</math>.
|alt1=Cartesian transport

|image2=Circle_transport.gif
|width2=200
|caption2=This transport is given by the metric <math>dr^2 + d\theta^2</math>.
|alt2=Polar transport
}}

Warning: This is parallel transport on the punctured plane ''along'' the unit circle, not parallel transport ''on'' the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

=== Riemann curvature tensor ===
{{Main|Riemann curvature tensor}}

The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map.{{sfn|Lee|2018|p=201}} The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.{{sfn|Lee|2018|p=200}}

Fix a connection <math>\nabla</math> on <math>M</math>. The ''[[Riemann curvature tensor]]'' is the map <math>R : \mathfrak X(M) \times \mathfrak X(M) \times \mathfrak X(M) \to \mathfrak X(M)</math> defined by
:<math>R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z</math>
where <math>[X, Y]</math> is the [[Lie bracket of vector fields]]. The Riemann curvature tensor is a <math>(1,3)</math>-tensor field.{{sfn|Lee|2018|pp=196–197}}

=== Ricci curvature tensor ===
{{Main|Ricci curvature}}

Fix a connection <math>\nabla</math> on <math>M</math>. The '''[[Ricci curvature tensor]]''' is
: <math>Ric(X,Y) = \operatorname{tr}(Z \mapsto R(Z,X)Y)</math>
where <math>\operatorname{tr}</math> is the trace. The Ricci curvature tensor is a covariant 2-tensor field.{{sfn|Lee|2018|p=207}}

==== Einstein manifolds ====
{{Main|Einstein manifold}}

The Ricci curvature tensor <math>Ric</math> plays a defining role in the theory of [[Einstein manifold]]s, which has applications to the study of [[gravity]]. A (pseudo-)Riemannian metric <math>g</math> is called an ''Einstein metric'' if ''Einstein's equation''
: <math>Ric = \lambda g</math> for some constant <math>\lambda</math>
holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called an ''Einstein manifold''.{{sfn|Lee|2018|p=210}} Examples of Einstein manifolds include Euclidean space, the <math>n</math>-sphere, hyperbolic space, and [[complex projective space]] with the [[Fubini-Study metric]].

=== Scalar curvature ===
{{Main|Scalar curvature}}