# Ngô Quốc Anh

## November 16, 2009

### R-G: Levi-Civita connection

Filed under: Riemannian geometry — Ngô Quốc Anh @ 2:39

Suppose $M$ is a differentiable manifold of dimension $n$.

Connection on vector bundles

Definition 1. A connection on a vector bundle $E$ is a map

$D : \Gamma(E) \to \Gamma(T^\star(M) \otimes E)$

which satisfies the following conditions

• For any $s_1, s_2 \in \Gamma(E)$, $D(s_1+s_2)=Ds_1 + Ds_2$.
• For $s \in \Gamma(E)$ and any $\alpha \in C^\infty(M)$, $D(\alpha s)=d\alpha \otimes s + \alpha Ds$.

If $X$ is a tangent vector field on $M$ (i.e. a section of the tangent bundle $TM$) one can define a covariant derivative along $X$, denoted by $D_X$, as follows

${D_X}s = \left\langle {X,Ds} \right\rangle$

where $\left\langle \cdot, \cdot \right\rangle$ represents the pairing between $TM$ and $T^\star M$.

Locally, a connection is given by a set of differential 1-forms. Suppose $U$ is a coordinate neighborhood of $M$ with local coordinates $x^i$, $1 \leq i \leq n$. Choose $q$ smooth sections $s_\alpha$ of $E$ on $U$ such that they are linearly independent everywhere. Such a set of $q$ sections is called a local frame field of $E$ on $U$. It is obvious that at every point $P \in U$

$\displaystyle \{ dx^i \otimes s_\alpha, 1 \leq i \leq n, 1 \leq \alpha \leq q\}$

forms a basis for the tensor space $T_P^\star \otimes E_P$. Because $Ds_\alpha$ is a local section on $U$, we can write

$\displaystyle D{s_\alpha } = \Gamma _{\alpha i}^\beta d{x^i} \otimes {s_\beta }$

where $\Gamma_{\alpha i}^\beta$ are smooth functions on $U$. Denote $\omega _\alpha ^\beta = \Gamma _{\alpha i}^\beta d{x^i}$ then $D{s_\alpha } = \omega _\alpha ^\beta \otimes {s_\beta }$.

Definition 2 (curvature operator). Suppose $X, Y$ are two arbitrary smooth tangent vector fields on the manifold $M$. Then

$\displaystyle R(X, Y) = D_XD_Y - D_YD_X - D_{[X,Y]}$

is the curvature operator of the connection $D$.

Obviously, $R(X,Y)$ has the following properties

• $R(X,Y)=-R(Y,X)$,
• $R(fX, Y)=f \cdot R(X,Y)$,
• $R(X,Y)(fs)=f \cdot (R(X,Y)s)$,

where $X, Y \in \Gamma(TM)$, $f \in C^\infty(M)$ and $s \in \Gamma(E)$.

Connection on tangent bundles (affine connections)

A tangent bundle $TM$ is an $n$-dimensional vector bundle determined intrinsically by the differentiable structure of an $n$-dimensional smooth manifold $M$. A connection of $TM$ is called an affine connection on $M$. Affine connection is usually denoted by $\nabla$.

Definition 3 (curvature tensor). The curvature tensor is a (1,3)-tensor defined by

$R(X,Y)Z = D_XD_YZ - D_YD_XZ - D_{[X,Y]}Z$.

In local coordinates, the curvature tensor is given by

$R = R_{ikl}^j\dfrac{\partial }{{\partial {x^j}}} \otimes d{x^i} \otimes d{x^k} \otimes d{x^l}$.

A simple calculation shows us that

$\displaystyle R_{ikl}^j = \left\langle {R\left( {\frac{\partial }{{\partial {x^k}}},\frac{\partial }{{\partial {x^l}}}} \right)\frac{\partial }{{\partial {x^i}}},d{x^j}} \right\rangle$.

Definition 4 (torsion tensor). The torsion tensor is a (1,2)-tensor defined by

$T(X,Y) = D_XY - D_YX - [X,Y]$.

In local coordinates, the torsion tensor is given by

$\displaystyle T = T_{ij}^k\frac{\partial }{{\partial {x^k}}}\otimes d{x^i} \otimes d{x^j}$.

A simple calculation shows us that

$\displaystyle T_{ij}^k = \Gamma _{ji}^k - \Gamma _{ij}^k$.

Definition 5 (torsion free). If the torsion tensor of an affine connection $D$ is zero, then the connection is said to be torsion free.

When $M$ is a Riemannian manifold with metric $g$ then we have the following definition

Definition 6 (Levi-Civita connection). An affine connection $\nabla$ is called a Levi-Civita connection if:

• It preserves the metric, i.e., for any vector fields $X$, $Y$, $Z$ we have

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

where $X(g(Y,Z))$ denotes the derivative of the function $g(Y,Z)$ along the vector field $X$.

• It is torsion free.

The first condition above is called metric connection condition. Thus, the Levi-Civita connection is the torsion free metric connection, i.e., the torsion free connection on the tangent bundle (an affine connection) preserving a given Riemannian metric.

There is a theorem in the literature saying that the Levi-Civita connection is unique and it is given by the following identity

$\displaystyle g({\nabla _X}Y,W) = \frac{1}{2}\left( {X(g(Y,W)) + Y(g(X,W)) - W(g(X,Y)) + g([X,Y],W) + g([W,X],Y) - g([Y,W],X)} \right)$.

In local coordinate, the Levi-Civita connection $\nabla$ is given by

$\displaystyle {\nabla _{\frac{\partial }{{\partial {x^i}}}}}\frac{\partial }{{\partial {x^j}}} = \Gamma _{ij}^k\frac{\partial }{{\partial {x^k}}}$

where $\Gamma _{ij}^k$ are called Christoffel symbols which are determined by

$\displaystyle \Gamma _{ij}^k = \frac{1}{2}{g^{kl}}\left( {{g_{il,j}} + {g_{jl,i}} - {g_{ij,l}}} \right)$

where ${g_{,m}} = \frac{{\partial g}}{{\partial {x^m}}}$.

Filed under: Riemannian geometry — Ngô Quốc Anh @ 0:20

Let’s start with a differentiable manifold M of dimension $n$. Throughout this topic, we denote by $P$ a point on $M$ and $(M,\varphi)$ its local chart (at $P$). A point $P$ is determined by $\varphi(P)$ hence it is often identified with $\varphi(P)$. We usually denote by $\varphi(P)=\{ x^i\} \in \mathbb R^n$ the local coordinates of $P$.

Definition 1. A tangent vector at $P$ is a map $X : f \mapsto X(f) \in \mathbb R$ defined on the set of the differentiable functions in a neighborhood of $P$, where $X$ satisfies the following conditions

• $X$ is linear, that is to say: if $\lambda, \mu \in \mathbb R$, then $X(\lambda f + \mu g)=\lambda X(f) + \mu X(g)$.
• $X(f)=0$ if $f$ is flat at $P$, i.e. $d(f \circ \varphi^{-1})=0$ at $\varphi(P)$.
• $X(fg)=f(P)X(g)+g(P)X(f)$.

Definition 2. The tangent space $T_P(M)$ at $P$ is the set of tangent vectors at $P$.

From the definition 1, let us show that the tangent space of definition 2 has a natural vector space structure of dimension $n$. We set

$(X+Y)(f) = X(f)+Y(f)$ and $(\lambda X)(f)=\lambda X(f)$.

With this sum and this product, $T_P(M)$ is a vector space. And now let us exhibit a basis. It is reasonable to define the tangent vector $\dfrac{\partial}{\partial x^i}$ at $P$. Precisely,

Definition 3. The tangent vector $\dfrac{\partial}{\partial x^i}$ at $P$ is defined to be

$\displaystyle\frac{\partial }{{\partial {x^i}}}\left( f \right) = \left( {\frac{\partial }{{\partial {x^i}}}\left( {f \circ {\varphi ^{ - 1}}} \right)} \right){\bigg|_{\varphi (P)}}$.

The vectors $\dfrac{\partial}{\partial x^i}$ are independent and they form a basis for $T_P(M)$. We usually call $\dfrac{\partial f}{\partial x^i}$ the directional derivative of $f$ in the direction $x^i$. For an arbitrary vector $X$, one can define the directional derivative of $f$ in the direction $X$ as following

$\displaystyle{\partial _X}(f) = X(f) = {X^i}\frac{{\partial f}}{{\partial {x^i}}}$

where $X^i$ denotes the $i$-th component of $X$ in this coordinate chart.

We now assume further that $(M, g)$ is a Riemannian manifold where $g$ is its metric. We are now in a position to define gradient for a smooth function.

Definition 4. For any smooth function $f$ on a Riemannian manifold $(M, g)$, the gradient of $f$ is the vector field $\nabla f$ such that for any vector field $X$,

$\displaystyle g(\nabla f,X) = {\partial _X}f$,   i.e.   $\displaystyle {g_P}({(\nabla f)_P},{X_P}) = ({\partial _X}f)(P)$

where $g_P(\cdot, \cdot)$ denotes the inner product of tangent vectors at $P$ defined by the metric $g$.

We now express the local form of the gradient at $P$. By definition 4, one has in the local coordinates

$\displaystyle {g_P}({(\nabla f)_P},{X_P}) = {X^i}\frac{{\partial f}}{{\partial {x^i}}}$.

If we assume $\displaystyle {(\nabla f)_P} = {Y^i}\frac{\partial }{{\partial {x^i}}}$ we then have

$\displaystyle {g_P}({(\nabla f)_P},{X_P}) = {g_P}\left( {{Y^i}\frac{\partial }{{\partial {x^i}}},{X^j}\frac{\partial }{{\partial {x^j}}}} \right) = {Y^i}{X^j}{g_{ij}}$.

Thus

$\displaystyle {Y^i}{X^j}{g_{ij}} = {X^i}\frac{{\partial f}}{{\partial {x^i}}}$

which implies after multiplying both sides by the matrix $(g^{ij})$

$\displaystyle {Y^i} = {g^{ij}}\frac{{\partial f}}{{\partial {x^j}}}$.

Therefore,

$\displaystyle {(\nabla f)_P} = {g^{ij}}\frac{{\partial f}}{{\partial {x^j}}}\frac{\partial }{{\partial {x^i}}}$.

We end this topic by showing what $|\nabla f|$ is? Roughly speaking, at a point $P$ since $\nabla f$ is a vector, then $|\nabla f|$ is nothing but its magnitude. To be exact, one defines

$\displaystyle \left| {\nabla f} \right| = \sqrt {{g_{ij}}{Y^i}{Y^j}} = \sqrt {{g_{ij}}\left( {{g^{im}}\frac{{\partial f}}{{\partial {x^m}}}} \right)\left( {{g^{jn}}\frac{{\partial f}}{{\partial {x^n}}}} \right)} = \sqrt {{g^{mn}}\frac{{\partial f}}{{\partial {x^m}}}\frac{{\partial f}}{{\partial {x^n}}}}$.

In Riemannian geometry, the lower index means differentiation and the upper index means component, therefore, we usually use $f_k$ to denote the quantity $\frac{\partial f}{\partial x^k}$. With this notation, $\left| {\nabla f} \right|=\sqrt{g^{mn}f_mf_n}$.

of $f$ in the direction $X$