Levi-Civita connection

In Riemannian geometry, the **Levi-Civita connection** is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.

## History

The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.

The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding

M

n

⊂

R

n
(
n
1
)

2

,

{\displaystyle M^{n}\subset \mathbf {R} ^{\frac {n(n+1)}{2}},}

since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.

### Remark

In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of
a space of constant curvature. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space. In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results. In the same year, Hermann Weyl generalized
Levi-Civita's results.

## Notation

* (*M*, *g*) denotes a Riemannian or pseudo-Riemannian manifold.
* *TM* is the tangent bundle of *M*.
* *g* is the Riemannian or pseudo-Riemannian metric of *M*.
* *X*, *Y*, *Z* are smooth vector fields on *M*, i. e. smooth sections of *TM*.
* [*X*, *Y*] is the Lie bracket of *X* and *Y*. It is again a smooth vector field.

The metric *g* can take up to two vectors or vector fields *X*, *Y* as arguments. In the former case the output is a number, the (pseudo-)inner product of *X* and *Y*. In the latter case, the inner product of *X**p*, *Y**p* is taken at all points *p* on the manifold so that *g*(*X*, *Y*) defines a smooth function on *M*. Vector fields act as differential operators on smooth functions. In a basis, the action reads

X
# f

X

i

∂

∂

x

i

# f

X

i

∂

i

f

{\displaystyle Xf=X^{i}{\frac {\partial }{\partial x^{i}}}f=X^{i}\partial \_{i}f}

where Einstein's summation convention is used.

## Formal definition

An affine connection ∇ is called a Levi-Civita connection if

1. *it preserves the metric*, i.e., ∇*g* = 0.
2. *it is torsion-free*, i.e., for any vector fields *X* and *Y* we have ∇*X**Y* − ∇*Y**X* = [*X*, *Y*], where [*X*, *Y*] is the Lie bracket of the vector fields *X* and *Y*.

Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text.

If a Levi-Civita connection exists, it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor *g* we find:

X

(

g
(
Y
,
Z
)

)

Y

(

g
(
Z
,
X
)

)

−
Z

(

g
(
Y
,
X
)

)

=
g
(

∇

X

Y

∇

Y

X
,
Z
)
g
(

∇

X

Z
−

∇

Z

X
,
Y
)
g
(

∇

Y

Z
−

∇

Z

Y
,
X
)
.

{\displaystyle X{\bigl (}g(Y,Z){\bigr )}+Y{\bigl (}g(Z,X){\bigr )}-Z{\bigl (}g(Y,X){\bigr )}=g(\nabla \_{X}Y+\nabla \_{Y}X,Z)+g(\nabla \_{X}Z-\nabla \_{Z}X,Y)+g(\nabla \_{Y}Z-\nabla \_{Z}Y,X).}

By condition 2, the right hand side is equal to

2
g
(

∇

X

Y
,
Z
)
−
g

(

[
X
,
Y
]
,
Z

)

g

(

[
X
,
Z
]
,
Y

)

g

(

[
Y
,
Z
]
,
X

)

,

{\displaystyle 2g(\nabla \_{X}Y,Z)-g{\bigl (}[X,Y],Z{\bigr )}+g{\bigl (}[X,Z],Y{\bigr )}+g{\bigl (}[Y,Z],X{\bigr )},}

so we find the Koszul formula

g
(

∇

X

Y
,
Z
# )

1
2

{

X

(

g
(
Y
,
Z
)

)

Y

(

g
(
Z
,
X
)

)

−
Z

(

g
(
X
,
Y
)

)

g

(

[
X
,
Y
]
,
Z

)

−
g

(

[
Y
,
Z
]
,
X

)

−
g

(

[
X
,
Z
]
,
Y

)

}

.

{\displaystyle g(\nabla \_{X}Y,Z)={\tfrac {1}{2}}{\Big \{}X{\bigl (}g(Y,Z){\bigr )}+Y{\bigl (}g(Z,X){\bigr )}-Z{\bigl (}g(X,Y){\bigr )}+g{\bigl (}[X,Y],Z{\bigr )}-g{\bigl (}[Y,Z],X{\bigr )}-g{\bigl (}[X,Z],Y{\bigr )}{\Big \}}.}

Since *Z* is arbitrary, this uniquely determines ∇*X**Y*. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi-Civita connection.

## Christoffel symbols

Let ∇ be the connection of the Riemannian metric. Choose local coordinates *x*1 … *xn* and let Γ*ljk* be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry

Γ

j
k

l

=

Γ

k
j

l

.

{\displaystyle \Gamma \_{jk}^{l}=\Gamma \_{kj}^{l}.}

The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

Γ

j
k

l

=

1
2

g

l
r

(

∂

k

g

r
j

∂

j

g

r
k

−

∂

r

g

j
k

)

{\displaystyle \Gamma \_{jk}^{l}={\tfrac {1}{2}}g^{lr}\left(\partial \_{k}g\_{rj}+\partial \_{j}g\_{rk}-\partial \_{r}g\_{jk}\right)}

where as usual *gij* are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix (*gkl*).

## Derivative along curve

The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by *D*.

Given a smooth curve *γ* on (*M*, *g*) and a vector field *V* along *γ* its derivative is defined by

D

t

# V

∇

γ
˙

(
t
)

V
.

{\displaystyle D\_{t}V=\nabla \_{{\dot {\gamma }}(t)}V.}

Formally, *D* is the pullback connection *γ*\*∇ on the pullback bundle *γ*\**TM*.

In particular,

γ
˙

(
t
)

{\displaystyle {\dot {\gamma }}(t)}

 is a vector field along the curve *γ* itself. If

∇

γ
˙

(
t
)

γ
˙

(
t
)

{\displaystyle \nabla \_{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)}

 vanishes, the curve is called a geodesic of the covariant derivative. Formally, the condition can be restated as the vanishing of the pullback connection applied to

γ
˙

{\displaystyle {\dot {\gamma }}}

:

(

γ

∗

∇

)

γ
˙

≡

{\displaystyle \left(\gamma ^{\*}\nabla \right){\dot {\gamma }}\equiv 0.}

If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.

## Parallel transport

In general, parallel transport along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.

The images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in polar coordinates. The metric of left image corresponds to the standard Euclidean metric

d

s

2

=
d

x

2

d

y

2

=
d

r

2

r

2

d

θ

2

{\displaystyle ds^{2}=dx^{2}+dy^{2}=dr^{2}+r^{2}d\theta ^{2}}

, while the metric on the right has standard form in polar coordinates, and thus preserves the vector

∂

∂
θ

{\displaystyle {\partial \over \partial \theta }}

 tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates:

d
# r

x
d
x
y
d
y

x

2

y

2

{\displaystyle dr={\frac {xdx+ydy}{\sqrt {x^{2}+y^{2}}}}}

d
# θ

x
d
y
−
y
d
x

x

2

y

2

{\displaystyle d\theta ={\frac {xdy-ydx}{x^{2}+y^{2}}}}

d

r

2

d

θ

2

=

(
x
d
x
y
d
y

)

2

x

2

y

2

(
x
d
y
−
y
d
x

)

2

(

x

2

y

2

)

2

{\displaystyle dr^{2}+d\theta ^{2}={\frac {(xdx+ydy)^{2}}{x^{2}+y^{2}}}+{\frac {(xdy-ydx)^{2}}{(x^{2}+y^{2})^{2}}}}

## Example: the unit sphere in **R**3

Let ⟨ , ⟩ be the usual scalar product on **R**3. Let **S**2 be the unit sphere in **R**3. The tangent space to **S**2 at a point *m* is naturally identified with the vector subspace of **R**3 consisting of all vectors orthogonal to *m*. It follows that a vector field *Y* on **S**2 can be seen as a map *Y* : **S**2 → **R**3, which satisfies

⟨

Y
(
m
)
,
m

⟩

=
0
,

∀
m
∈

S

2

.

{\displaystyle {\bigl \langle }Y(m),m{\bigr \rangle }=0,\qquad \forall m\in \mathbf {S} ^{2}.}

Denote as *dmY*(*X*) the covariant derivative of the map *Y* in the direction of the vector *X*. Then we have:

**Lemma:** The formula

(

∇

X

Y

)

(
m
# )

d

m

Y
(
X
)
⟨
X
(
m
)
,
Y
(
m
)
⟩
m

{\displaystyle \left(\nabla \_{X}Y\right)(m)=d\_{m}Y(X)+\langle X(m),Y(m)\rangle m}

defines an affine connection on **S**2 with vanishing torsion.
**Proof:** It is straightforward to prove that ∇ satisfies the Leibniz identity and is *C*∞(**S**2) linear in the first variable. It is also a straightforward computation to show that this connection is torsion free. So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all *m* in **S**2

⟨

(

∇

X

Y

)

(
m
)
,
m

⟩

=
0

(
1
)
.

{\displaystyle {\bigl \langle }\left(\nabla \_{X}Y\right)(m),m{\bigr \rangle }=0\qquad (1).}

Consider the map *f* that sends every *m* in **S**2 to ⟨*Y*(*m*), *m*⟩, which is always 0. The map *f* is constant, hence its differential vanishes. In particular

d

m

f
(
X
# )

⟨

d

m

Y
(
X
)
,
m

⟩

⟨

Y
(
m
)
,
X
(
m
)

⟩

=

{\displaystyle d\_{m}f(X)={\bigl \langle }d\_{m}Y(X),m{\bigr \rangle }+{\bigl \langle }Y(m),X(m){\bigr \rangle }=0.}

The equation (1) above follows. Q.E.D.
In fact, this connection is the Levi-Civita connection for the metric on **S**2 inherited from **R**3. Indeed, one can check that this connection preserves the metric.