Tangent space, Derivative

In the previous post we have seen some examples of manifolds and the most general ways in which they occur. By now we know what we mean by differentiable functions on the differentiable manifolds. But we have not defined what do we mean by derivative of a function. So the the natural way to do is to take a coordinate atlas and then define the derivative of the function to be the usual derivative in $\mathbb{R}^n$ under these coordinate charts. But the definition depends on the choice of coordinate charts. But the basic motivation behind studying this topic of differentiable manifolds is that the concept of derivative is something intrinsic and is independent of the choice of coordinates. Of course we could say say that take a coordinate charts and define and then say how the object changes with respect to coordinate changes and show that it changes consistently etc. But we can define in a very intrinsic manner. One should think the concept of linear transformation and different matrix representations as analogies. The matrix representation and choice of a basis give coordinates to some vectors ( geometric,intrinsic objects) but they have intrinsic meaning.

Lets get back to $\mathbb{R}^n$ and see what we mean by derivative. Lets us first consider functions to $\mathbb{R}$ . Then we can define the derivative as the limit of the newtons quotient. This tells you how the function changes when the moves from a point x to a very near point x+h in some direction h. So we to talk about derivative one has to tell how the function changes as one moves in some direction. Now moving to derivative of function to $\mathbb{R}^k$ one could see that the derivative is really a linear transformation on $\mathbb{R}^n \to \mathbb{R}^k$ . This say as you move from the point x in some direction ( vector in $\mathbb{R}^n$ ) the change will be from the point f(x) in a direction in $\mathbb{R}^k$ .The transformation is infact linear. Even in the 1-dimensional case R we have to think of the derivative as a transformation form $\mathbb{R}^n$ to $\mathbb{R}$ . We think of it as number. But this number is just a coordinate representation of the linear transformation. And we confuse the matrix $[f'(x)]$ with $f'(x)$ .

Now lets us see what we need to define the derivate in the case of a manifold. Imagine a manifold which is inside some $\mathbb{R}^n$ . Now we want to speak of the derivative of the function. So we should say how that function changes when we move to a very near by point in some direction. Now there are no such concepts of direction etc for a abstract manifold. But if it a manifold as above which sits in $\mathbb{R}^n$ we have a tangent hyperplane intersecting the manifold at a point. So we can speak of the function changing along those directions. But even in this case the tangent plane is an external object to the manifold. We have to define the direction or the tangent plane ,tangent vectors intrinsically in terms of the manifold. So what should we do?

Here is an idea! Just take the curves passing through the point. Consider the equivalence class of all the curves having same tangent vectors to be representing the tangent vector.But how do we define the tangent vectors for the curves. We don’t know what it means. I just restated what we wanted in a different manner. But we can define the tangent vector by going to the coordinates charts and defining tangent vector as derivative that point of the curve at that point and saying how it changes under coordiante transformations. ( Note that curves or more precisely oriented curves are function from an interval I in R to the manifold. So we know what derivatives means when we choose coordiante chart for the manifold x.)

But this again the old story. We used coordinate representation. Here is another idea. If we have a direction(tangent vector)in R^n then we can know how a function changes along that direction. So given any function we can compute the directional derivative. So we identify the direction with the directional derivative. The we think of the direction by how functions change in that direction. So we have a linear maps on functions which gives the directional derivative. We call this itself as the direction or the tangent vector. This linear has to satisfy the leibniz property: $v(fg)=f(x)v(g)+g(x)v(f)$ Here $v(f)$ is basically the directional derivative $D_v$ of a smooth function $f:\mathbb{R}^n \to \mathbb{R}$ . Such linear maps satisfying leibniz rule are called derivations.

We can add these linear maps to get another linear map and hence all these maps form a vector space called set of derivations. So what we are going to do is that we define the tangent space to be this vector space of linear maps which satisfy leibniz property for an abstract manifold.
So we take all the smooth functions on a smooth manifold and take the set of derivations at a point x to be the tangent space at x. In local coordinates these derivations exactly corresponds to the directional derivatives $d/dx1,d/dx2,...,d/dx_k$ at x. So the tangent space is a vector space of dimension n. So it matches with our previous requirements.The tangent space at a point x in $\mathbb{R}^n$ will be latex $\mathbb{R}^n$ .

Note that we have defined the tangent space for a smooth manifold as set of derivations. Similarly we may want to define the tangent space for any $C^r$ manifold as the set of derivations. But the problem is that the this set is a infinite dimensional space. It doesnt match with our requiremnts. So what we do is to take a subspace of it spanned by some n vectors which are$latex d/dx1,d/dx2,…,d/dx_k$ in some coordinate charts. We can prove such a thing is well defined i.e, there is consistency under coordinate changes.(Note that any general tangent vector will be of the form $\sum f_i(x_1,x_2,...,x_n)d/dx_i$

On the whole the basic idea was to think of tangent directions to be given by the directional derivatives. Once we define this notion its all normal once again. We just look at the tangent vectors as we used to see, in terms of pictures as given by tangent planes etc. Now we define derivative of a function. The derivative $df|_x$ at a point x of a function is an linear map from the tangent space at x of X to tangent space at $f(x)$ of the codomain such that $df|_x(v)=v(f)$ .

So we have now defined derivative.The basic idea is that it acts on the tangent vectors near x to tangent vectors near f(x) linearly.

Implicit function theorem: In the previous post we stated this theorem. We say more about the manifolds we get as level sets. That is we can know that tangent space at point of this space form the derivative map of the original function. Let $p$ be in $L_c(f) \subset X^{n+k}$ (a point in the level set at a regular value c). Let $df:T_p(X) \to T_{f(p)}(Y)$ be the derivative map at p. Now the tangent space at the point p is the nullspace of this map. It just says that along the directions on the level set space, the function f doesnt change. Hence the tangent space exactly corresponds to these directions i.e., the nullspace of f.

Example: We shall compute the derivative of the function $f:X \to X^3$ from the space $M_n(R)$ to $M_n(R)$ . Note that $M_n(R)$ can be looked at as $n^2$ dimensional manifold (Isomorphic to $\mathbb{R}^{n^2})$ .Now the tangent space at any point $X_0$ is also $n^2$ dimensional. Infact it is again $M_n(R)$ .Now the derivative $df$ is given by $df(Y)=\lim_{t\to 0}frac{(X_0+tY)^3-(X_0)^3}{t}$ i.e., direction derivative along tangent vector Y. There $df(Y) =X_0^2Y+X_0YX_0+YX_0^2$ . At along $I$ it is $3X_0^2$ . So the derivative $3X^2$ of $X^3$ is to be interpreted as a directional derivative along the unit vector 1 in the case of functions on the real line. Note that such computations are important in Lie theory.

So we are done with basic definitions!!! We know what a differentiable manifold is,what a tangent space is and what a derivative is. Thus we setup a category of differentiable manifolds. The objects being differentiable manifolds, morphisms(maps) being differentiable maps between the manifolds.The isomorphisms are diffeomorphisms. So we can try to understand the category. We can ask for classifications. We can ask how different are the categories of $C^r$ manifolds $r=0,1,2,3..\infty,\omega$ from each other. We can ask embedding questions. We can try to understand the automorphism groups of different objects. We try to see how many different differentiable manifolds are homeomorphic i.e., they are same as topological spaces. So we can try to understand how different are these differentiable structures of a same manifold. We can try to see how the algebraic tools used to understand the topological spaces can be used here.(Algebraic tools are obviously the most powerful tools we have to understand these spaces!) But before going to all that stuff we shall see some critical point theory. These are important from a practical point of view because it is at these points smooth functions attain extremal values.

Share this:

Related

Leave a comment Cancel reply