Ngô Quốc Anh

November 16, 2009

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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