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}}}.

R-G: Tangent space, gradient

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

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