Differential Geometry – Second Fundamental Form of General Immersion

Question

I'm working on evolution equations of mean curvature flow, and to take an initial step, I guess I would have to know the second fundamental form anyway. Thank you.

In literature, the mean curvature flow is often present as a one-parameter family of immersions $F:M^n\times[0,T)\to(N^{n+1},\bar{g})$ subject to the differential equation
$$\frac{\partial F}{\partial t}=H\nu.$$
The $n$-dimensional manifold $M$ is not necessarily a subset of the Riemannian $(n+1)$-manifold $N$ and is sometimes called a hypersurface in $N$ via the time-dependent immersion $F_t:=F(\cdot,t)$. If we endow $M$ with the (time-dependent) pull-back metric $g:=F_t^*\bar{g}$, then a Levi-Civita connection $\nabla$ will be induced accordingly ($\overline{\nabla}$ is reserved for the Levi-Civita connection on $N$). Now, to define the vector-valued second fundamental form $\mathbf A$ of the hypersurface $M$, I figure that a comparison between these two connections is necessary. In a Riemannian geometry book by John Lee, this is a task that seems natural to me, because Lee defines submanifolds as subsets of ambient manifolds and therefore considers the inclusion map $\iota:M\hookrightarrow\widetilde{M}$. And what we do in this book is define $\mathbf A$ by
$$\widetilde{\nabla}_X Y=\nabla_X Y+\mathbf{A}(X,Y),\tag{1}$$
where $X,Y$ are vector fields on $M$ and the LHS is interpreted in terms of $X,Y$ being extended to vector fields on neighborhoods of $M$ in $\widetilde{M}$.

Now I'm wondering if there is a way to generalize the preceding concept to the domain $M$ of $F_t$ (rather than the image $M_t:=F_t(M)$). I mean, is there an equation that is similar to (1) and goes like
$$\overline{\nabla}_X Y=\nabla_X Y+\mathbf{A}(X,Y)?\tag{2}$$
I don't know how to imagine the decomposition of $\overline{\nabla}_X Y$ because, in contrast to (1), I don't know how to decompose $\overline{\nabla}_X Y$ into a normal component plus a tangential component pictorially.

On the other hand, I saw a definition of the scalar-valued second fundamental form elsewhere which reads:

If $\nu$ is a local choice of unit normal for $F_t(M)$, we often work in an adapted orthonormal frame $e_1,\ldots,e_n,\nu$ in a neighborhood of $F_t(M)$ such that $e_1|_p,\ldots,e_n|_t\in T_p M\subseteq T_{F_t(p)}N$ for $p\in M$. Then the second fundamental form $A=(h_{ij})$ is given by
$$h_{ij}=\langle\bar{\nabla}_{e_i}\nu,e_j\rangle=-\langle\nu,\bar{\nabla}_{e_i}e_j\rangle.\tag{3}$$

I was wondering the intended meaning of (3). At first glance, it does not make sense to me because, to my knowledge, second fundamental forms are meant to be a geometric comparison between $M$ and $N$, and the map $F_t$ linking these two manifolds does not show its appearance in (3). I was thinking, could (3) be understood as
$$h_{ij}=\color{red}{?}=-\left<\nu,\frac{\partial^2 F_t}{\partial e_i\partial e_j}\right>_\bar{g}?\tag{4}$$
I was thinking about (4) because the book Lecture Notes on Mean Curvature Flow by Carlo Mantegazza includes a similar expression. In this book, the author considers the immersion $F_t$ with $N$ superseded by $\mathbb{R}^{n+1}$, and then defines $\mathbf{A}$ to be $h_{ij}\nu$ with
$$h_{ij}=\left<\nu,\frac{\partial^2 F_t}{\partial x_i\partial x_j}\right>,$$
the scalar-valued second fundamental form. But how could we define a unit normal $\nu$ along $M$ without $M$ sitting inside an ambient manifold?

Can someone tell me how to define the second fundamental form of the immersion $F_t:M\to(N,\bar{g})$? Specifically, can someone tell me how to compare the geometry of the domain $M$ with that of the codomain $N$? Textbooks that expand on this topic are welcome. Thank you.

Remark. If you traced the questions I've asked so far, you would probably notice that I keep circling back to the same problem. I guess that's because I never understand the true meaning of the second fundamental form for an arbitrary immersion, and now I want to fix it once and for all. As indicated previously, I did learn a basic knowledge of Riemannian submanifolds in John Lee's book, including his definition of second fundamental forms, but those discussions are based on a genuine subset $M$ of a Riemannian manifold $(N,\bar g)$, which is seldom the case in a research paper. Lee does mention the more general case of an isometric immersion on page 226 of his IRM, but I doubt if I understand his intended meaning correctly:

Choose an appropriate neighborhood $U$ in $M$ so that $F_t$ restricts to an embedding on $U$. Then define, exactly in the subset sense, the second fundamental form $\mathrm{II}_U$ between $F_t(U)$ and $N$. If we can show the behavior of $\mathrm{II}_U$ is independent of the choice of $U$, we will succeed in defining the second fundamental form between $M$ and $N$, even if $M$ may not be a subset of $N$. (3) is perhaps a local representation of $\mathrm{II}_U$. Did I get it?

You probably wonder why I focus my attention on the domain $M$. I'd tell you that's because I saw many authors identify this $M$ as a hypersurface in $N$. Of course, it sounds weird to me because $M$ sometimes does not sit in $N$ as a subset. But that will be fine if we manage to define the second fundamental form between $M$ and $N$, since we are to call $M$ a hypersurface in $N$.

Answer 1

The idea of transport of structure is a very common and recurring idea in math. For example, if $V$ is a vector space, and $W$ is a set and $f:V\to W$ is a bijection, then there is a unique way to endow the set $W$ with operations of addition and scalar multiplication such that $f:V\to W$ becomes a linear isomorphism (simply define $w_1+_Ww_1:= f(f^{-1}(w_1)+_Vf^{-1}(w_2))$ and $c\cdot_W w:=f(c\cdot_Vf^{-1}(w))$). Or for example, if $f:X\to Y$ is a bijection and $X$ is a topological space, then there is a unique topology on $Y$ such that $f$ becomes a homeomorphism. And so on.

The shortest possible answer is that locally an immersion is an embedding, so locally you can transport structure (Riemannian metrics, connections, etc) from the domain to the image/vice-versa by composing or by pushing-forward or by pulling back the map in question. Also, definitions of curvature and second-fundamental forms are local in nature, which is why everyone writes out the details in the case of an embedded Riemannian submanifold $M\subset N$, but then says in a line or two that everything carries over to the more general case of isometric immersions $f:M\to N$, modulo some suppressions in the notation (i.e identifying $T_pM$ with its isometric image $Tf_p[T_pM]\subset T_{f(p)}N$, and abusing (well… just suppressing) notation by writing $(T_pM)^{\perp}$ when really one means $\left(Tf_p[T_pM]\right)^{\perp}$, the orthogonal complement taken in $T_{f(p)}N$).

Anyway, unwinding explicitly what all this transport of structure entails, we have the following. Let $M,N$ be Riemannian manifolds and $f:M\to N$ an isometric immersion. Let $\nabla^M,\nabla^N$ be their respective Levi-Civita connections. Our goal is to define a bilinear (symmetric) vector bundle morphism $A:TM\oplus TM\to TN$ over the map $f$, which takes values in ‘$(TM)^{\perp}$’. More explicitly, we want a smooth, fiberwise bilinear map $A$ which for each $x\in M$ and tangent vectors $\xi_x,\eta_x\in T_xM$ gives us a vector $A(\xi_x,\eta_x)\in \bigg(Tf_x[T_xM]\bigg)^{\perp}\subset T_{f(x)}N$. To define this, fix $x\in M,\xi_x,\eta_x\in T_{x}M$.

• Consider any smooth vector field $Y$ on $N$ such that $Y_{f(x)}=Tf_{x}(\eta_x)\in T_{f(x)}N$ (i.e we’re fixing the value of the vector field $Y$ at one point $f(x)$).
• Given the vector field $Y$ on $N$ above, define a vector field $\tilde{Y}$ on $M$ by setting for all $p\in M$, $\tilde{Y}(p):=(Tf_p)^{-1}\bigg(\left(Y_{f(p)}\right)_{\parallel_{p}}\bigg)$, where $\parallel_p$ denotes the orthogonal projection $T_{f(p)}N\to Tf_p[T_pM]$. In the subset case, this corresponds to the obvious parallel projection tangent to $M$ once you restrict the field to $M$. Note, we used that $f$ is an immersion here so that we could take the inverse of $Tf_p:T_pM\to Tf_p[T_pM]\subset T_{f(p)}N$.
• Then, consider the difference $\nabla^N_{Tf_{x}(\xi_x)}Y- Tf_{x_0}\left(\nabla^M_{\xi_x}\tilde{Y}\right)\in T_{f(x)}N$. Recall that connections only need a single tangent vector in their lower slot, so this expression makes sense.

The claim (which I won’t prove) is then that if we define $A(\xi_x,\eta_x)$ to be the expression in the third bullet point, then this value in $T_{f(x)}N$ does not depend on the choice $Y$ made in the first bullet point, and furthermore, this expression is symmetric. Hence, we get our desired symmetric bilinear bundle morphism $A:TM\oplus TM\to TN$ over the base map $f$, which particularly takes values ‘orthogonal to $M$ in $N$’. It is in this sense that the second fundamental form measures the extrinsic geometry of $M$ once it is isometrically immersed in $N$. This is the sort of thing you write out once and then never again, because honestly, all this notation isn’t that enlightening (the second paragraph justifies why we can treat $f$ as if it is an inclusion map).

Finally, regarding the meaning of $\nu$ is that it is smooth map $\nu:M\to TN$ such that for each $x\in M$, $\nu(x)\in \bigg(Tf_x[T_xM]\bigg)^{\perp}\subset T_{f(x)}N$. I’ll leave it to you to now properly interpret the formulae, and insert the $f$ in appropriate places.

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Lastly, let’s go with the unnecessarily general and abstract nuclear option (I’m not even sure if these ramblings will be more helpful or harmful, but here it is regardless). Here is a 3 step process; the key to making all this ‘rigorous’ is the notion of pullback vector bundle, as it allows you to compare things over different base spaces $M$ vs $N$ (for example, this is also what you use when talking about covariant derivatives along curves, especially to make sense of things like the LHS of the geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$).

• First, the ‘general’ setting of second fundamental forms is as follows (see Ivo Terek’s notes for a very nice presentation along these lines (among other nice stuff)). The idea is we have a vector bundle $(X,\pi,M)$ and a connection $D$ on $X$. Suppose also that we have a direct sum bundle decomposition $X=L\oplus L^{\perp}$ (I’m using $\perp$ purely symbolically… there’s no need for any metrics at this stage). This induces two connections $\nabla$ on $L$ and $\nabla^{\perp}$ on $L^{\perp}$ by projection. These two connections can be put back together via (internal) direct sum $\nabla\oplus\nabla^{\perp}$ to get a connection on $L\oplus L’=X$. So, we now have two connections on $X$. The first is our original $D$, and the second is our sum of projected connections $\nabla\oplus \nabla^{\perp}$. The difference of these two connections $\alpha:=D-\nabla\oplus\nabla^{\perp}$ will be a ‘tensor’, more precisely it is a bundle map $\alpha:TM\to\text{End}(X)$, or equivalently, a bilinear bundle map $\alpha:TM\oplus X\to X$. This is the second-fundamental form/shape-tensor of the connection $D$ relative to the subbundle decomposition $X=L\oplus L^{\perp}$ (i.e it measures the geometry of how the two subbundles $L,L^{\perp}$ are sitting inside the big bundle $X$, when measured using the connection $D$). In fact, since $\alpha:TM\to \text{End}(X)$ is endomorphism-valued, one can use the direct sum decomposition $X=L\oplus L^{\perp}$ to give it a $2\times 2$ block decomposition, and one can show (see the notes) that the block form is $\alpha=\begin{pmatrix}0&B\\ A&0\end{pmatrix}$, where $A:TM\to \text{Hom}(L,L^{\perp})$ and $B:TM\to \text{Hom}(L^{\perp},L)$. In the case when you have a bundle metric and a metric-compatible connection and $L,L^{\perp}$ really are orthogonal complements, you can show (see the notes) that $B=-A^*$ i.e $A,B$ are negative metric-adjoints of each other. This is why in Riemannian geometry, one always focuses only on the $A$ term, $A:TM\to\text{Hom}(L,L^{\perp})$, or equivalently a bilinear $A:TM\oplus L\to L^{\perp}$.

The Riemannian geometry special case is that you have a Riemannian manifold $N$, a submanifold $M\subset N$, and you set $X=(TN)|_M$ the restricted tangent bundle of $N$ over the base $M$, and $L=TM$ and $L^{\perp}=(TM)^{\perp}$ is the normal bundle. This coincides perfectly with what you know about the (vector) second fundamental form: it takes in two tangent vectors to the submanifold and produces something normal $A:TM\oplus TM\to (TM)^{\perp}$.

• Usually what happens is we have a vector bundle $(Y,\pi,N)$, and a smooth map $f:M\to N$, and we would really like to work over the base space $M$. Hence, we work with the pullback vector bundle $(X=f^*Y,\pi_f,M)$ over $M$. Next, if we have a connection $D$ initially on $Y$, then this gets pulled back to a connection $f^*D\equiv D^f$ on $Y$, satisfying some natural compatibility conditions.
• We put the previous two ideas together. Start with a vector bundle $(Y,\pi,N)$ and a connection $D$ on $Y$. Then, consider a map $f:M\to N$. Pull everything back to $M$. So we now have the vector bundle $(X,\pi,M)$, a connection $D^f$ on $X=f^*Y$. Suppose now further that we have a direct sum decomposition $X=L\oplus L^{\perp}$ (this really just amounts to having at each $x\in M$, a direct sum decomposition of $Y_{f(x)}$, which is canonically isomorphic to $X_x$, into $L_x\oplus L_x^{\perp}$, i.e a direct sum decomposition in the original bundle $Y$, but ‘along $f$’). The connection $D^f$ projects to two connections $\nabla^f,\nabla^{f,\perp}$ in $L,L^{\perp}\subset X=f^*Y$ respectively, and next, let $\alpha^f:=D^f-\nabla^{f}\oplus\nabla^{f,\perp}$ be the second fundamental form. In particular, for any section $\psi$ of the subbundle $L$, any point $x\in M$ and any $h_x\in T_xM$, we have \begin{align} \alpha(h_x,\psi(x))&=D^f_{h_x}\psi-(\nabla^f\oplus\nabla^{f,\perp})_{h_x}(\psi)\\ &=D^f_{h_x}\psi-[\nabla^f_{h_x}\psi+0] \tag{since $\psi$ has no $L^{\perp}$ component}\\ &=D^f_{h_x}\psi-(D^f_{h_x}\psi)_{\parallel}\\ &=(D^f_{h_x}\psi)_{\perp}, \end{align} where $\parallel,\perp$ are the direct sum projections $X\to L$ and $X\to L^{\perp}$.