Tangent Vectors (on Manifolds)

In mathematics, a tangent generally has to do with an entity touching something at only one point, and representing the rate of change of something — a curve, a surface, a function, whatever. For example, a tangent to a curve is a straight line touching the curve at only one point, and whose gradient is equal to the gradient (=rate of change) of the curve at that point. Here I will try to explain the idea of tangent vectors on manifolds.

The most intuitive way to visualise them is as little tangents to curves on a manifold. A curve on a manifold is some function from an interval of the real numbers to a subset of the manifold: think of taking a straight line (representing the interval of real numbers) and twisting it whichever way you want (excluding crossings with itself) to fit somewhere on the manifold. The tangent vectors at a point on the manifold are simply the “lines” that (locally) touch the curve on the manifold at only that point. The curve itself exists independently of whatever local coordinate system we choose to use on the manifold. Since the tangent vectors are simply auxiliary characteristics of the curve, we would expect the tangent vectors to also exist independently of local coordinate systems — that is, while their labels in different coordinate systems may be different, the labels must be related to each other in a determined way, so that we know two different labels are in fact referring to the same object. In mathematical parlance, this translates to the tangent vector being subject to the chain rule for transformations between coordinates. If V is one coordinate patch with coordinates x^i_V and U is another patch with coordinates x^i_U, and the point p_0 lies on their overlap, then the chain rule says the following:
\left(\frac{dx_V^i}{dt}\right)_0 = \sum^n_{j=1}\frac{\partial x_V^i}{\partial x^j_U}\left(p_0\right) \left(\frac{dx_U^j}{dt}\right)_0

\left(\frac{dx_V^i}{dt}\right)_0 and \left(\frac{dx_U^j}{dt}\right)_0 are the tangent vectors to the curve at p_0 under coordinate systems x_V and x_U respectively. So all that scary equation says is that tangent vectors must obey the chain rule under transformations between overlapping coordinate systems.

We can even start off defining tangent vectors to be just those entities that transform that way. Here is Frankel [1]:

Definition: A tangent vector, or contravariant vector, or simply a vector at p_0 \in M^n, call it X, assigns to each coordinate patch (U, x) holding p_0, an n-tuple of real numbers (X_U^i) = (X^1_U, \dots , X_U^n) such that if p_0 \in U \cap V, then X_V^i = \sum_j \left(\frac{\partial x_V^i}{\partial x_U^j}\left(p_0\right)\right) X^j_U

Thus we need not be restricted to tangent lines to curves on the manifold: this definition lets us include any vector entity that happens to follow the transformation rule between two coordinate systems as a tangent vector. \frac{dx_U^j}{dt} and \frac{dx_V^i}{dt} in the ‘curve’ definition of a tangent vector are replaced by X_U^j and X_V^i in the ‘general’ definition. The ‘curve’ definition, though, is still a better way to visualise tangent vectors, since it offers at least a picture of tiny little tangent lines nibbling at a curve on a manifold.

[1] Frankel, T. The Geometry of Physics. Cambridge: Cambridge University Press, 1997.

Published in:  on September 2, 2007 at 6:45 am Leave a Comment