Lecture 3: Tangent Vectors and the Tangent Space

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Lecture 2 described scalar fields on manifolds. We now turn to vectors on smooth manifolds. We must first talk about smooth paths in M.

Definition 3.1 A smooth path in the smooth manifold M is a smooth map defined on an open segment of the real line, r: (-a, a) M, where r is a vector-valued function with coordinates (y1, y2, . . ., ys).

We say that r is a smooth path through m M if r(t0) = m for some t0.

We can specify a path in M at m by its coordinates:

where m is the point (y1(t0), y2(t0), . . . , ys(t0)). Equivalently, since the ambient and local coordinates are functions of each other, we can also express a path--at least that part of it inside a coordinate neighborhood--in terms of its local coordinates:

Examples 3.2
(a) (borrowed from Example 1.7 in Lecture 1) Straight lines in E3
(b) (from Example 1.7 in Lecture 1) Helix in a cylinder radius r embedded in E3
(c) A smooth path in Sn

Definition 3.3 A tangent vector at m M Er is a vector v in Er of the form v = y'(t0) for some path y = y(t) in M through m (with y(t0) = m.

Examples 3.4
(a) Let M be the surface y3 = y12 + y22, which we paramaterize by

This corresponds to the single chart (U=M; x1, x2), where

To specify a tangent vector, let us first specify a path in M, such as

(Check that the equation of the surface is satisfied.) This gives the path shown in the figure.
Now we obtain a tangent vector field along the path by taking the derivative:

(To get actual tangent vectors at points in M, evaluate this at a fixed point t0.)

Note We can also express the coordinates xi in terms of t:

giving

since xi = yi for this manifold. We also think of this as the tangent vector, given in terms of the local coordinates. A lot more will be said about the relationship between the above two forms of the tangent vector below.

Algebra of Tangent Vectors: Addition and Scalar Multiplication

The sum of two tangent vectors is, geometrically, also a tangent vector, and the same goes for scalar multiples of tangent vectors. However, we have defined tangent vectors using paths in M, and we cannot produce these new vectors by simply adding or scalar-multiplying the corresponding paths: if y = f(t) and y = g(t) are two paths through m M where f(t0) = g(t0) = m, then adding them coordinate-wise need not produce a path in M. However, we can add these paths using some chart as follows.

Choose a chart x at m, with the property (for convenience) that x(m) = 0. Then the paths x(f(t)) and x(g(t)) (defined as in the note above) give two paths through the origin in coordinate space. Now we can add these paths or multiply them by a scalar without leaving coordinate space and then use the chart map to lift the result back up to M. In other words, define

(f+g)(t) = x-1(x(f(t)) + x(g(t))
and(f)(t) = x-1(x(f(t))).
Taking their derivatives at the point t0 will, by the chain rule, produce the sum and scalar multiples of the corresponding tangent vectors. Since we can add and scalar-multiply tangent vectors

Definition 3.5 If M is an n-dimensional manifold, and m M, then the tangent space at m is the set Tm of all tangent vectors at m.

The above constructions turn Tm into a vector space.

Let us return to the issue of the two ways of describing the coordinates of a tangent vector at a point m M: writing the path as yi = yi(t) we get the ambient coordinates of the tangent vector:

y'(t0) =
dy1

dt
, ... ,
dys

dt
t=t0   Ambient coordinates
and, using some chart x at m, we get the local coordinates
x'(t0) =
dx1

dt
, ... ,
dxn

dt
t=t0   Local coordinates

Question In general, how are the dxi/dt related to the dyi/dt?

Answer By the chain rule,

and similarly for dy2/dt and dy3/dt. Thus, we can recover the original three ambient vector coordinates from the local coordinates. In other words, the local vector coordinates completely specify the tangent vector.

Note The chain rule as used above shows us how to convert local coordinates to ambient coordinates and vice-versa:

Converting Between Local and Ambient Coordinates of a Tangent Vector

If the tangent vector V has ambient coordinates (v1, v2, . . . , vs) and local coordinates (v1, v2, . . . , vn), then they are related by the formulae

vi= n

k=1
yi

xk
vk
and
vi= s

k=1
xi

yk
vk.

Note To obtain the coordinates of sums or scalar multiples of tangent vectors, simply take the corresponding sums and scalar multiples of the coordinates. In other words:

(v+w)i = vi + wi and (v)i = vI
just as we would expect to do for ambient coordinates. (Why can we do this?)

From now on, we shall omit the summation signs, and use the Einstein Summation Convention:

Einstein Summation Convention

If an index appears twice in an expression, then summation over that index is implied.

Thus,

n

k=1
yi

xk
vk   becomes  
yi

xk
vk  (because the index k repeats)
and
s

k=1
xi

yk
vk   becomes  
xi

yk
vk   (again because the index k repeats).

Examples 3.4 Contd.
(b) Take M = En, and let v be any vector in the usual sense with coordinates i. Choose x to be the usual chart xi = yi. If p = (p1, p2, . . . , pn) is a point in M, then v is the derivative of the path

x1 = p1 + t1
x2 = p2 + t2;
    . . .
xn = pn + tn
at t = 0. Thus this vector has local and ambient coordinates equal to each other, and equal to which are the same as the original coordinates. In other words, the tangent vectors are "the same" as ordinary vectors in En.

(c) Let M = S2, and the path in S2 given by

This is a path (circle) through m = (0, 0, 1) following the line of longitude x2 = 0, and has tangent vector

(c) We can also use the local coordinates to describe a path; for instance, the path in part (b) can be described using spherical polar coordinates by

The derivative

give the local coordinates (the coordinates of its image in coordinate Euclidean space).

(e) In general, if (U; x1, x2, . . . , xn) is a coordinate system near m, then we can obtain paths yi(t) by setting

where the constants are chosen to make xi(t0) correspond to m. (The paths in parts (c) and (d) are examples of this.) To view this as a path in M, we just apply the parametric equations yi = yi(xj), giving the yi as functions of t.

The associated tangent vector at the point where t = t0 is called /xi. It has local coordinates

ij is called the Kronecker Delta, and is defined by

Question Which matrix has ij entry equal to ij?
Answer

Question Using the Einstein summation convention, evaluate viij.
Answer

We can now get the ambient coordinates by the above conversion formula (we are using the Einstein summation convention from this point on):

We call this vector /xi. Summarizing,

Definition of


xi



xi
is the vector whose local coordinates are given by

    j th coordinate=


    xi
    j
     
     
    =ij =
    xj

    xi
    .

Its ambient coordinatres are given by

    j th coordinate=
    yj

    xi
    .

What do these strange vectors look like?

Now that we have a better feel for local and ambeinet coordinates of vectors, let us state some more "general nonsense": Let M be an n-dimensional manifold, and let m M.

Proposition 3.6 (The Tangent Space)

There is a linear one-to-one correspondence between tangent vectors at m and plain old vectors in En. In other words, the tangent space "looks like" En.

Click here for a proof (which will also explain why local coordinates are better than ambient ones).

Question Wait a minute! Isn't that obvious from the picture? The tangent space is just an n-dimensional plane, and all n-dimensional planes are just copies of n-dimensional space!

Answer Geometrically, that seems true -- at least in three dimensions. But remember, we have defined the tangent space at m M as the set of tangent vectors to paths through m. How can we be so sure that this 1-1 correspondence works (a) with this definition, and (b) in arbitrary s-dimensional space? That is why we really need the proof.

Note Under the one-to-one correspondence in the proposition, the standard basis vectors in En correspond to the tangent vectors /x1, /x2, . . . , /xn. Therefore, the latter vectors are a basis of the tangent space Tm.


Exercise Set 3

1. Suppose that v is a tangent vector at m M with the property that there exists a local coordinate system xi at m with vi = 0 for every i. Show that v has zero coordinates in every coefficient system, and that, in fact, v = 0.

2. (a) Calculate the ambient coordinates of the vectors / and / at a general point on S2, where and are spherical polar coordinates ( = x1, = x2).
(b) Sketch these vectors at some point on the sphere.

3. Prove that

4. Consider the torus T2 with the chart x given by

y1 = (a+b cos x1)cos x2
y2 = (a+b cos x1)sin x2
y3 = b sin x1
with 0 < xi < 2. Find the ambeint coordinates of the two orthogonal tangent vectors at a general point, and sketch the resulting vectors.

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Last Updated: January, 2002
Copyright © Stefan Waner