Fdtxjr

Let $f: M^nto mathbbS^n+p$ be an isometric immersion. The cone over $f$ is defined to be the immersion
beginalign*
F:mathbbR_+times M &to mathbbR^n+p+1\
(t,x)&mapsto tf(x)
endalign*
Compute the second fundamental form of $F$.

[where $mathbbR_+:=tinmathbbRmid t>0$]

I could verify that $F$ is an immersion by considering $f(x)=(f_1(x),...,f_n+p+1(x))inmathbbR^n+p+1$ with $sum_if_i^2=1$. That way, $langlefracpartial fpartial x_i,frangle=0$ for all $i$ and, since $f$ is an immersion, we have $fracpartial fpartial x_1,...,fracpartial fpartial x_n$ linearly independent, so $textrank(dF)=textrankleft(f,fracpartial fpartial x_1,...,fracpartial fpartial x_nright)=n+1$.

For the $2^textnd$ fundamental form, here's where I'm at: If $widetildenabla$, $nabla$ are Levi-Civita connections for $mathbbR^n+p+1$ and $mathbbR_+times M$ respectively, the second fundamental form is by definition $alpha(X,Y)=widetildenabla_widetildeXwidetildeY-nabla_XY$. By tensoriality and symmetry of $alpha$, we only need to compute $alphaleft(fracpartialpartial t,fracpartialpartial tright),alphaleft(fracpartialpartial t,fracpartialpartial x_iright)$ and $alphaleft(fracpartialpartial x_i,fracpartialpartial x_jright)$.

To compute $alphaleft(fracpartialpartial t,fracpartialpartial tright)$ for example, can see that $nabla_fracpartialpartial tfracpartialpartial t=0$, but I don't know how compute $widetildenabla_fracpartial Fpartial tfracpartial Fpartial t$. Obviously, $fracpartial Fpartial t(t,x)=f(x)$, but I can't figure out what $widetildenabla_f(x)f(x)$ even means.

Any suggestions?

The question is purely local, so we may choose a coordinate chart $varphi colon U subset mathbbR^n to M$ around some point $x in M$. We use this chart to indentify points in $U$ and $varphi(U)$, and thus we regard $f$ locally as $varphi circ f$, so that the mapping $f colon U to mathbbR^n+p+1$ is a parametrization of its image, and
$$
langle f, f rangle = 1
$$

The mapping
$F colon mathbbR_+ times U to mathbbR^n+p+1 colon (t, x) mapsto t f(x)$
is also a local parametrization of its image, which is a piece of the cone $C = F(mathbbR_+ times U)$.

We calculate at the point $(t, x) in C$.

Here is a picture showing schematically what is going on:

Corresponding to the chart $varphi$, there are local coordinates $x^i$ in $U$, and let $t$ be a local coordinate in $mathbbR_+$, so that $ t, x^i, i=1, dots, n $ form a coordinate system on $mathbbR_+ times U$.

Let us introduce the following notation: $partial_i := fracpartialpartial x^i$, $partial_t := fracpartialpartial x^i$, $f_i := partial_i f$, $F_i := partial_i F$, $F_t := partial_t F$, $i = 1, dots, n$. Since $F = t f$, we have:
$$
F_i = tf_i text and F_t = f
$$
Notice that $F_i$ and $F_t$ are vector fields along the cone $C$, and
$$
F_i|_(t,x) = t pmatrix
f^1_i(x)\
vdots\
f^n+p+1_i(x)

text and
F_t|_(t,x) = pmatrix
f^1(x)\
vdots\
f^n+p+1(x)

$$

In order to differentiate $F_i$ and $F_t$ in $mathbbR^n+p+1$ we need some local coordinates there and some extensions $widetildeF_i$ and $widetildeF_t$ of those fields, so we just take the slice coordinates in $mathbbR_+ times U times mathbbR^p$
$$
y^0 = t, y^1 = x^1, dots, y^n = x^n, y^n+1 = y^n+1, dots, y^n+p = y^n+p
$$
(where $y^n+1, dots, y^n+p$ are the standard coordinates in $mathbbR^p$), and define
$$
widetildeF(t, x^1, dots, x^n, *, dots, *) := F(t, x^1, dots, x^n)
$$
so that $widetildeF_i = partial_i widetildeF$ and $widetildeF_t = partial_i widetildeF$.
Since in $mathbbR^n+p+1$ we have the standard Euclidean metric and the standard Euclidean connection (all Christoffel symbols vanish identically),
we have
$$
widetildenabla_i widetildeF_j = partial_i F_j \
widetildenabla_t widetildeF_i = partial_t F_i \
widetildenabla_t widetildeF_t = partial_t F_t
$$

In order to compute the s.f.f. $alpha(X,Y)=widetildenabla_widetildeXwidetildeY-nabla_XY$ it remains to understand what $nabla_XY$ is here.

It is known that $nabla_XY = (widetildenabla_widetildeXwidetildeY)^top = Pi(widetildenabla_widetildeXwidetildeY)$, where the tangential projection operator is given by
$$
Pi colon v mapsto frac1^2 langle v, F_t rangle F_t + sum_i=1^n frac1^2 langle v, F_i rangle F_i
$$
This is almost it, because now we can write
$$
nabla_i F_j = Pi(widetildenabla_i widetildeF_j) = frac1^2 langle widetildenabla_i widetildeF_j, F_t rangle F_t + sum_k=1^n frac1 langle widetildenabla_i widetildeF_j, F_k rangle F_k \
nabla_t F_i = Pi(widetildenabla_t widetildeF_i) = frac1^2 langle widetildenabla_t widetildeF_i, F_t rangle F_t + sum_k=1^n frac1 langle widetildenabla_t widetildeF_i, F_k rangle F_k \
nabla_t F_t = Pi(widetildenabla_t widetildeF_t) = frac1^2 langle widetildenabla_t widetildeF_t, F_t rangle F_t + sum_k=1^n frac1 langle widetildenabla_t widetildeF_t, F_k rangle F_k
$$
and collecting all the pieces it is now straightforward to obtain the final expressions.