Volume Comparison Notes

The goal of this final project is to demonstrate the relationship between the curvature of a Riemannian manifold (a local property) and its global topology by presenting Myer’s theorem as a corollary of Bishop’s volume comparison theorem. Along the way, we will derive the first and second variational formulas for area for Riemannian manifolds which will be used to develop the volume comparison results. In the last section, we will mention further results connecting the geometry and curvature of a Riemannian manifold to its topology that can be arrived at via the framework of volume comparison. The statements and proofs of these results are based on the exposition in the first two chapters of Peter Li’s Geometric Analysis ((Li 2012)).

1.1 Basic Concepts from Riemanian Manifolds

We first introduce basic concepts of Riemannian manifolds that we will use throughout. The purpose of this section is more to establish notation and standard results that we will use in Sections 2 and 3 rather than exposition. To shorten the exposition, we will just state these definitions and results below without proof. A useful resource for this material is John M. Lee’s Riemannian Manifolds: An Introduction to Curvature ((Lee 1997)). We endeavor to give explicit citations and page numbers of major named results in this section (such as the statement of the Hopf-Rinow Theorem), but otherwise you can assume that all basic definitions and facts here are taken from (Lee 1997). Note that (Lee 1997) also gives proofs of Bonnet’s and Myer’s theorems for Riemannian manifolds, though more directly using Jacobi fields rather than via volume comparison (see p.200-201 of (Lee 1997)).

Let \(M\) be an \(m\)-dimensional smooth manifold. Then we can make endow \(M\) with a Riemannian metric which is a choice of positive definite inner product, denoted \(g\) on the tangent spaces \(T_pM\) that smoothly vary with \(p\) (in other words, it is a 2-tensor on the tangent bundle \(TM\)). If \(X, Y\) are two vector fields on \(M\), then their inner product is denoted

If \(\{x_1,\dots,x_m\}\) are local coordinates at \(p\) and \(\{\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_m}\}\) are the corresponding coordinate vector fields, then we also denote

\[ g_{ij} = g\left(\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\right) = \left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j} \right\rangle. \]

Then \(g_{ij}\) can be thought of as an \(m\times m\) matrix that varies with choice of point \(p \in M\) (so we might also denote the Riemannian metric by \(g_{ij}(p)\) with the vector fields above evaluated at \(p\)). Recall from a previous homework problem in this course that we can use a partition of unity argument to show the existence of Riemannian metrics on smooth manifolds.

Let \(\mathscr{T}(M)\) be the space of smooth vector fields on \(M\). Then we have a notion of covariant derivatives in the notion of a connection \(\nabla: \mathscr{T}(M)\times\mathscr{T}(M)\to\mathscr{T}(M)\) which satisfies the following: let \(a, b \in \mathbb{R}, f, g \in C^\infty(M), X, Y, Z \in \mathscr{T}(M)\) then

We can have two additional properties of metric compatibility and torsion free.

Then requiring the connection to additionally be metric compatible and torsion free makes \(\nabla\) into the unique Levi-Civita connection (also called the Riemannian connection) on \((M, g)\) ((Lee 1997), p.68).

A curve \(\gamma: I\subseteq\mathbb{R} \to M\) is called a geodesic if its acceleration with respect to \(\nabla\) is 0, that is, \(D_t \dot{\gamma} \equiv 0\) where \(D_t\) is the covariant derivative operator induced from the connection. We saw the exponential map on smooth manifolds in class, much of which carries over into the Riemannian setting. Additionally, we will also use the notion of geodesic balls in a Riemannian manifold which is the image of Euclidean balls in the tangent spaces under the exponential map as well as the notion of normal coordinates. For the sake of brevity, we do not state these results but rather refer the reader to (Lee 1997), p. 72-81 for the general theory.

\[ \nabla_{\frac{\partial}{\partial x_i}}\frac{\partial}{\partial x_j} = \Gamma^k_{ij} \frac{\partial}{\partial x_k}. \]

The Riemannian metric \(g\) on \(M\) induces a distance function \(d_g\) on \(M\) as the infimum over the length of piecewise smooth curves connecting \(p\) to \(q\) in \(M\). Then this turns \(M\) into a bona fide metric space with the topology induced by the metric coinciding with the original manifold topology of \(M\). Then the Hopf-Rinow Theorem ((Lee 1997), p.108) states that a connected Riemannian manifold is geodesically complete (i.e. all maximal geodeics are defined on all of \(\mathbb{R}\)) if and only if it is complete as a metric space. One particular corollary of the Hopf-Rinow Theorem that we will rely on later is that \(M\) is complete if and only if any two points in \(M\) can be joined by a minimizing geodesic segment ((Lee 1997), p.111). The diameter of \(M\) is then defined as \[\text{diam}(M) = \sup\{d_g(p,q):p,q\in M\}.\]

\[ R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X, Y]} Z, \quad X,Y,Z \in \mathscr{T}(M) \]

and satisfies the following identities (collected here for the sake of convenience) ((Li 2012), p.2):

\[ \begin{aligned} &R(X,Y)Z = -R(Y,X) Z \\ &R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0 \\ &\langle R(X,Y)Z, W \rangle = \langle R(Z,W)X, Y \rangle \\ &R(fX,Y)Z = R(X,fY)Z = R(X,Y)(fZ) = fR(X,Y)Z \end{aligned} \]

We can then define the other notions of curvature on \(M\) from the Riemannian curvature tensor. For any 2-dimensional subspace \(\sigma_p\) of \(T_pM\) spanned by orthonormal vectors \(X, Y\), then the sectional curvature of \(\sigma_p\) at \(p\) is given by

whenever \(\{e_1,\dots,e_m\}\) is an orthonormal basis of \(T_pM\). In particular, the diagonal elements are

If \(N\) is an \(n\)-dimensional submanifold of \(M\) with \(n < m\), then for smooth vector fields \(X, Y \in \mathscr{T}(M)\), then given the Levi-Civita connection \(\nabla\) on \(M\), we can define \(\nabla^t\) on \(N\) by

where the superscript \(t\) on the right-hand side denotes taking the component of \(\nabla_XY\) tangential to \(N\). Then it is easy enough to check that \(\nabla^t\) is still a connection on \(N\) and is in fact the Levi-Civita connection on \(N\) with respect to the metric induced from \(g\) on \(M\).

We similarly define the Second Fundamental Form of \(N\) by taking the negative of the normal component of \(\nabla\):

Finally we define the mean curvature vector on \(N\) by taking the trace of the second fundamental form

where \(\{e_1,\dots,e_n\}\) is an orthonormal basis of \(T_pN\). Note that while the definitions of the second fundamental form and mean curvature vector can be found in (Lee 1997), p.134 and p.142 respectively, the notation and definitions above more closely follow (Li 2012), p.2 for the sake of keeping the notation consistent with the proofs in Sections 2 and 3 below (in particular, the definition of the second fundamental form given above differs from that given in (Lee 1997) by a negative sign).

2 Variational Formulas for Area

In this section, we derive the first and second variational formulas for area for a Riemmanian manifold. The proofs and derivations in this section and Section 3 below are from chapters 1 and 2 of (Li 2012).

2.1 First Variational Formula for Area

As in the previous section, let \(N\) be an \(n\)-dimensional submanifold of an \(m\)-dimensional manifold \(M\) with \(n < m\). Given \(\varepsilon > 0\), we can view a smooth map

as giving a one-parameter family of deformations, or variations of \(N\). That is, we have a family of manifolds parameterized by the “time” variable \(t \in (-\varepsilon,\varepsilon)\) given by \(N_t = \phi(N,t)\) with \(N = N_0 = \phi(N, 0)\). Then for \(\{x_1,\dots,x_n\}\) a local coordinate system around a point \(p \in N\), we can denote by \(\{x_1,\dots,x_n,t\}\) to be a local coordinate system around \((p, 0) \in N \times (-\varepsilon,\varepsilon)\). For convenience, we will use \(\partial_i\) to denote \(\frac{\partial}{\partial x_i}\) for \(i = 1,\dots,n\) and \(\partial_t\) to denote \(\frac{\partial}{\partial t}\). Then we also denote \[e_i = d\phi(\partial_i),\quad T = d\phi(\partial_t).\] We denote by \(g_{ij} = \langle e_i,e_j\rangle\) to be the metric on \(N_t\) induced from the metric on \(M\) (understanding \(e_{n+1}\) to be \(T\) in this case).

Let \(dA_t\) be the area element of \(N_t\) with respect to the induced metric and we write \(dA_t = J(p,t)dA_0\) for \((p,t)\in N\times(-\varepsilon,\varepsilon)\) and for \(t\) sufficiently close to 0. Then we have \[J(p,t) = \frac{\sqrt{\det(g_{ij}(p,t))}}{\sqrt{\det(g_{ij}(p,0))}}.\] Then our goal will be to compute \[J'(p,t) = \frac{\partial J(p,t)}{\partial_t}.\]

If we let \(\{x_1,\dots,x_n\}\) to be normal coordinates around \(p\in N\), then we have that \(g_{ij}(p,0) = \delta_{ij}\) the kronecker delta, and additionally since in these coordinates the Christoffel symbols vanish at \((p, 0)\), we also have that \(\nabla_{e_i}e_j(p,0) = 0\). Then in these coordinates, since \(g_{ij}(p,0) = \delta_{ij}\) we have that \(\det(g_{ij}(p,0)) = \det\delta_{ij} = 1\) and \[\begin{aligned} J'(p,0) &= \left.\frac{\partial}{\partial_t}J(p,0)\right|_{t=0} = \left.\frac{\partial}{\partial_t}\frac{\sqrt{\det(g_{ij}(p,t))}}{\sqrt{\det(g_{ij}(p,0))}}\right|_{t=0} = \left.\frac{\partial}{\partial_t}\sqrt{\det(g_{ij}(p,t))}\right|_{t=0} \\ &= \left.\frac{1}{2}g(p,t)^{-\frac{1}{2}}\right|_{t=0}\left.\frac{\partial}{\partial_t}g(p,t)\right|_{t=0} = \frac{1}{2}g'(p,0).\end{aligned}\] On the other hand, by the Laplace/cofactor expansion for the determinant, we have that \[\det(g_{ij}) = \sum\limits_{j=1}^n g_{1j}c_{1j}\] where \(c_{ij}\) are the cofactors of \(g_{ij}\). So by the product rule, we have that \[\begin{aligned} g'(p,0) &= \left(\sum\limits_{j=1}^n g_{1j}(p,0)c_{1j}(p,0)\right)' \\ &= \sum\limits_{j=1}^n g'_{1j}(p,0)c_{1j}(p,0) + \sum\limits_{j=1}^n g_{1j}(p,0)c'_{1j}(p,0).\end{aligned}\] Note that in normal coordinates since \(g_{ij} = \delta_{ij}\), \(g_{1j}(p,0)\) is 1 if \(j = 1\) and 0 otherwise, so the last sum becomes just \(c'_{11}(p,0)\). Similarly in the first sum when \(j = 1\), the \(1,1\)-th minor of \(g_{ij} = \delta_{ij}\) is still \(\delta_{ij}\) (one dimension lower), so \(c_{11}(p,0) = 1\) while when \(j \neq 1\), the \(1,j\)-th minor will have a 0 on the main diagonal, so \(c_{1j}(p,0) = 0\), so the first sum becomes just \(g'_{11}(p,0)\). So we have that \[g'(p,0) = g'_{11}(p,0) + c'_{11}(p,0).\] Then continuing the calculation on \(c'_{11}\) we eventually get that \[g'(p,0) = \sum\limits_{i=1}^n g'_{ii}(p,0).\]

Now we evaulate \(g'_{ii}(p,0)\): We have \[g'_{ii}(p,0) = T\langle e_i,e_i\rangle = \langle\nabla_T e_i,e_i\rangle + \langle e_i,\nabla_T e_i\rangle = 2\langle \nabla_T e_i, e_i \rangle = 2\langle \nabla_{e_i}T, e_i\rangle.\] Note that since both \(T\) and \(e_i\) are coordinate vector fields, their Lie bracket \([T, e_i] = 0\) so by the torsion free property of the connection, we can interchange their order when taking the covariant derivative in the last equality above.

Note that \(g'(p,0)\) is independent of choice of basis and hence is defined globally. Then decomposing \(T = T^t + T^n\) into tangential and normal components and by metric compatibility, we have that \[\begin{aligned} \sum\limits_{i=1}^n \langle \nabla_{e_i}T, e_i\rangle &= \sum\limits_{i=1}^n \langle \nabla_{e_i}T^t, e_i \rangle + \sum\limits_{i=1}^n \langle\nabla_{e_i}T^n,e_i\rangle \\ &= \sum\limits_{i=1}^n \langle\nabla_{e_i}T^t, e_i\rangle + \sum\limits_{i=1}^n\left(e_i\langle T^n, e_i\rangle - \langle T^n, \nabla_{e_i}e_i\rangle\right) \\ &= \sum\limits_{i=1}^n \langle\nabla_{e_i}T^t, e_i\rangle - \sum\limits_{i=1}^n \langle T^n, \nabla_{e_i}e_i\rangle \\ &= \text{div}T^t + \langle T^n, \vec{H}\rangle ,\end{aligned}\] where we have used in the second-to-last equality the fact that the \(e_i\)’s are all tangential so each \(\langle T^n, e_i\rangle = 0\) and in the last equality the definition of the divergence operator and the definition of the mean curvature vector \(\vec{H}\).

So we have shown that \[\left.\frac{\partial J(p,t)}{\partial t}\right|_{(p,0)} = \frac{1}{2}g'(p,0) = \text{div}T^t + \langle T^n, \vec{H}\rangle.\] In other words, we have that the first variation for the volume form at the point \((p,0)\) is given by

\[ \frac{d}{dt} dA_t |_{(p,0)} = \left(\text{div}T^t + \langle T^n, \vec{H}\rangle\right) dA_0|_{(p,0)}. \](2.1)

However, the right hand side is intrinsically defined independent of choice of coordinates and hence Equation (2.1) is valid at any arbitrary point.

Moreover, if \(T\) is a compactly supported variational vector field on \(N\), then by the generalized divergence theorem we have that

\[ \left.\frac{d}{dt}A(N_t)\right|_{t=0} = \int_{N} \langle T^n, \vec{H}\rangle. \](2.2)

From Equation (2.2) above we can see that the mean curvature of \(N\) is identically 0 if and only if \(N\) is a critical point of the area functional. We call immersed submanifolds \(N \xhookrightarrow{} M\) minimal in this case.

2.2 Second Variational Formula for Area

For the second variational formula for area, we consider \[\phi: N\times(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon) \xhookrightarrow{} M\] a two-parameter family of variations of \(N\) with “time” variables \(t,s\). Then similarly to above we will denote for \(i = 1,\dots,n\) \[e_i = d\phi\left(\frac{\partial}{\partial x_i}\right),\quad T = d\phi\left(\frac{\partial}{\partial t}\right),\quad S = d\phi\left(\frac{\partial}{\partial s}\right)\] Then again, we will write \[dA_{t,s} = J(p,t,s)dA_{0,0}\] where \(dA_{t,s}\) is the area element of \(N_{t,s} = \phi(N, t, s)\) and \[J(p,t,s) = \frac{\sqrt{\det{g_{ij}}(p,t,s)}}{\sqrt{\det{g_{ij}(p,0,0)}}}.\] Then our goal will be to compute \(\frac{\partial^2}{\partial s\partial t}J(p,t,s)\)

Note that by a straightforward calculation (similarly to the one for the first variational formula for area above) we can find that \[\frac{\partial}{\partial t}J(p,t,s) = \sum\limits_{i,j=1}^n g^{ij}\langle\nabla_{e_i}T,e_j\rangle J(p,t,s),\] where \(g^{ij}\) denotes the inverse matrix of \(g_{ij}\).

Now differentiating the above with respect to \(s\), we have \[\begin{aligned} \frac{\partial^2 J}{\partial s \partial t} &= \sum\limits_{i,j=1}^n S\left(g^{ij}\langle\nabla_{e_i} T,e_j\rangle J\right) \\ &= \sum\limits_{ij=1}^n (Sg^{ij})\langle\nabla_{e_i}T,e_j\rangle J + \sum\limits_{ij=1}^n g^{ij} S\langle\nabla_{e_i}T,e_j\rangle J + \sum\limits_{ij=1}^n g^{ij}\langle\nabla_{e_i}T,e_j\rangle S(J).\end{aligned}\] Once again we choose normal coordinates and evaluate the above at \((p,0,0)\). Then in the above, \(J(p,0,0) = 1\) and each \(g^{ij} = \delta_{ij}\) the kronecker delta. So the above becomes

\[ \begin{aligned} \frac{\partial^2 J}{\partial s \partial t} &= \sum\limits_{ij=1}^n (Sg^{ij})\langle\nabla_{e_i}T,e_j\rangle + \sum\limits_{ij=1}^n \delta_{ij} S\langle\nabla_{e_i}T,e_j\rangle + \sum\limits_{ij=1}^n \delta_{ij}\langle\nabla_{e_i}T,e_j\rangle S(J) \nonumber \\ &= \sum\limits_{i,j=1}^n (Sg^{ij})\langle\nabla_{e_i}T,e_j\rangle + \sum\limits_{i=1}^n S\langle\nabla_{e_i}T,e_i\rangle + \sum\limits_{i,j=1}^n\delta_{ij}\langle\nabla_{e_i}T,e_j\rangle S(J). \end{aligned} \](2.3)

To simplify the first sum on the right-hand side above, recall that since \(g^{ij}, g_{ij}\) are inverse to each other, we have the equality \[\sum\limits_{k=1}^n g^{ik}g_{kj} = \delta_{ij}\] We can differentiate this with respect to \(S\) and compute

\[ \sum\limits_{k=1}^n(Sg^{ik})g_{kj} = -\sum\limits_{k=1}^ng^{ik}(Sg_{kj}) \](2.4)

So multiplying on the right by \(g^{\ell j}\) and renaming some indices, the left-hand side of Equation (2.4) above becomes \[\sum\limits_{k=1}^n(Sg^{ik})g_{k\ell}g^{\ell j} = \sum\limits_{k=1} (Sg^{ik})\delta_{kj} = Sg^{ij}\] and on the right-hand side we have \[\begin{aligned} -\sum\limits_{k,\ell=1}^n g^{ik}(Sg_{kj})g^{\ell j} &= -\sum\limits_{k,\ell=1}^n g^{ik}\left(S\langle e_k,e_j\rangle\right)g^{\ell j} \\ &= -\sum\limits_{k,\ell=1}^n g^{ik}\left(\langle\nabla_S e_k,e_j\rangle+\langle e_k,\nabla_S e_j\rangle\right)g^{\ell j} \\ &= - \langle\nabla_{e_i} S,e_j\rangle - \langle e_i,\nabla_{e_j} S\rangle\end{aligned}\] where in the last equality we have simplified by choosing normal coordinates and we can exchange the order in the covariant derivative since \(S\) and \(e_i\) are coordinate vector fields (as above). So after multiplying by \(g^{\ell j}\), (2.4) above becomes \[Sg^{ij} = -\langle \nabla_{e_i} S, e_j\rangle - \langle \nabla_{e_j} S, e_i\rangle.\] So the first sum in Equation (2.3) becomes

\[ \sum\limits_{i,j=1}^n (Sg^{ij})\langle\nabla_{e_i}T,e_j\rangle = -\sum\limits_{i,j=1}^n\langle\nabla_{e_i}S,e_j\rangle\langle\nabla_{e_i}T,e_j\rangle - \sum\limits_{i,j=1}^n \langle\nabla_{e_j}S,e_i\rangle\langle\nabla_{e_i}T,e_j\rangle. \](2.5)

To simplify the second sum in Equation (2.3) above, recall that \[R(S,e_i)T = \nabla_S\nabla_{e_i}T - \nabla_{e_i}\nabla_S T - \nabla_{[S,e_i]}T\] and since \(S, e_i\) are coordinate vector fields, their Lie bracket \([S, e_i] = 0\) so the last term above vanishes and we have \[\nabla_S\nabla_{e_i}T = R(S,e_i)T + \nabla_{e_i}\nabla_S T.\]

Hence, the second sum in Equation (2.3) can be written as \[ \begin{aligned} \sum\limits_{i=1}^n S\langle\nabla_{e_i}T,e_i\rangle &= \sum\limits_{i=1}^n \langle\nabla_S\nabla_{e_i}T,e_i\rangle + \sum\limits_{i=1}^n\langle\nabla_{e_i}T,\nabla_S e_i\rangle \nonumber \\ &= \sum\limits_{i=1}^n\langle R(S,e_i)T,e_i\rangle + \sum\limits_{i=1}^n \langle\nabla_{e_i}\nabla_S T,e_i\rangle + \sum\limits_{i=1}^n\langle\nabla_{e_i} T, \nabla_{e_i} S\rangle.\end{aligned} \](2.6)

To deal with the last sum above, consider that at \((p,0,0)\) and in normal coordinates, we have \[\begin{aligned} S(J)|_{(p,0,0)} &= \left.\frac{\partial}{\partial s}J(p,t,s)\right|_{(p,0,0)} \\ &= \left.\sum\limits_{i,j=1}^n g^{ij}\langle\nabla_{e_i}S,e_j\rangle J(p,t,s)\right|_{(p,0,0)} \\ &= \sum\limits_{i,j=1}^n\delta_{ij}\langle\nabla_{e_i}S,e_j\rangle J(p,0,0) \\ &= \sum\limits_{j=1}^n\langle\nabla_{e_j}S,e_j\rangle.\end{aligned}\] So the last sum in Equation (2.3) above becomes

\[ \sum\limits_{i,j=1}^n\delta_{ij}\langle\nabla_{e_i}T,e_j\rangle S(J) = \left(\sum\limits_{i=1}^n\langle\nabla_{e_i}T,e_i\rangle\right)\left(\sum\limits_{j=1}^n\langle\nabla_{e_j}S,e_j\rangle\right) \](2.7)

Substituting (2.5), (2.6) and (2.7) into Equation (2.3) yields the second variational formula for area:

\[ \begin{aligned} \frac{\partial^2 J}{\partial s \partial t} &= -\sum\limits_{i,j=1}^n \langle\nabla_{e_i} S,e_j\rangle\langle \nabla_{e_i}T,e_j\rangle - \sum\limits_{i,j=1}^n\langle\nabla_{e_j}S,e_i\rangle\langle\nabla_{e_i}T,e_j\rangle \nonumber \\ &\quad + \sum\limits_{i=1}^n\langle R(S,e_i)T,e_i\rangle + \sum\limits_{i=1}^n \langle\nabla_{e_i}\nabla_S T,e_i\rangle + \sum\limits_{i=1}^n\langle\nabla_{e_i}T,\nabla_{e_i}S\rangle \nonumber \\ &\quad + \left(\sum\limits_{i=1}^n\langle\nabla_{e_i}T,e_i\rangle\right)\left(\sum\limits_{j=1}^n\langle\nabla_{e_j}S,e_j\rangle\right). \end{aligned} \](2.8)

2.3 Special Case

While the two variational formulas arrived at in above are quite general, for the purposes of the next section we are especially interested in the following special case, which we simply state below: let \(N\) be an oriented hypersurface in an oriented manifold \(M\) (i.e. it has codimension 1) and the variation is restricted to be given by hypersurfaces which are constant distance from \(N\). The variational vector field in this case is given by \(e_m\) with \(\nabla_{e_m}e_m = 0\) identically. In this case, the mean curvature vector can be written \(\vec{H} = He_m\) and the two variational formulas for area become

\[ \frac{\partial^2}{\partial t^2}J(p,0) = -\sum\limits_{i,j = 1}^{m-1} h_{ij}^2(p)J(p,0) - R(e_m,e_m)(p)J(p,0) + H^2(p)J(p,0) \](2.10)

where \(h_{ij}\) are the components of the second fundamental form and \(R(X,X)\) denotes the Ricci curvature of \(M\) in the direction of a vector \(X\).

3 Volume Comparison Results

Using the special case for the first and second variational formulas for area stated at the end of the previous section, in this section we develop some volume comparison results for Riemannian manifolds.

3.1 Bishop Comparison Theorem

Let \(M\) be a complete \(m\)-dimensional Riemannian manifold and \(p \in M\) be a point. Then in polar normal coordinates at \(p\), we can write the volume element as \[J(\theta,r)dr \wedge d\theta,\] where \(d\theta\) is the area element of the unit \((m-1)\)-sphere. We will denote by \(B_p(r)\) to be the geodesic ball of radius \(r\) centred at \(p\) in \(M\) (that is, it is the image under the exponential map at \(p\) of the Euclidean ball of radius \(r\) in \(T_pM\)). Then by the Gauss Lemma ((Lee 1997), p.102), the area element of the boundary \(\partial B_p(r)\) is given by \(J(\theta, r)d\theta\). Let \(x \in (\theta, r)\) be a point not in the cut-locus of \(p\) (so geodesics are still minimizing), then writing Equations (2.9) and (2.10) in polar coordinates we have

\[ J''(\theta,r) = -\sum\limits_{i,j=1}^{m-1}h^2_{ij}(\theta,r)J(\theta,r)-R_{rr}J(\theta,r)+H^2(\theta,r)J(\theta,r) \](3.2)

where \(R_{rr} = R\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial r}\right)\) denotes the Ricci curvature in the radial direction, and \(H(\theta,r)\), and \(h_{ij}(\theta,r)\) are respectively the mean curvature and the second fundamental form of \(\partial B_p(r)\) at the point \(x = (\theta, r)\).

Note that by the Cauchy-Schwarz inequality we have \[\begin{aligned} \left(\sum\limits_{i=1}^{m-1}h_{ii}\right)^2 &= \left(\sum\limits_{i=1}^{m-1}h_{ii}\cdot 1\right)^2 \leq \left(\sum\limits_{i=1}^{m-1}h_{ii}^2\right)\left(\sum\limits_{i=1}^{m-1}1^2\right) = (m-1)\left(\sum\limits_{i=1}^{m-1}h_{ii}^2\right).\end{aligned}\] Hence, we have that \[\begin{aligned} \sum\limits_{i,j=1}^{m-1}h_{ij}^2 &= \sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}h_{ij}^2 \\ &\geq \sum\limits_{i=1}^{m-1}h_{ii}^2 \\ &\geq \frac{\left(\sum\limits_{i=1}^{m-1}h_{ii}\right)^2}{m-1} = \frac{H^2}{m-1}.\end{aligned}\]

Then substituting this inequality into Equation (3.2) and using Equation (3.1), we have

\[ \begin{aligned} J'' &= -\sum\limits_{i,j=1}^{m-1}h_{ii}^2J - R_{rr}J + H^2J \nonumber \\ &\leq \frac{H^2}{m-1}J - R_{rr}J + H^2J \nonumber \\ &= J\left(\frac{H^2}{m-1}-R_{rr}+H^2\right) \nonumber \\ &= \frac{m-2}{m-1}H^2J - R_{rr}J \nonumber \\ &= \frac{m-2}{m-1}(J')^2J^{-1} - R_{rr}J. \end{aligned} \](3.3)

Note that as \(r\to0\) we have the initial conditions \[\begin{aligned} &J(\theta,r) \sim r^{m-1} \\ &J'(\theta,r) \sim (m-1)r^{m-2}\end{aligned}\] since close to \(p\) the metric is locally Euclidean. Also note that if we choose \(M\) to be a simply connected manifold with constant sectional curvature \(K\) (which we will call a space form), then \(R_{rr} = (m-1)K\) and the inequality above in Equation (3.3) becomes an equality with \[J'' = \frac{m-2}{m-1}(J')^2J^{-1} - (m-1)KJ.\]

Theorem 1 (Bishop’s Comparison Theorem). Let \(M\) be an \(m\)-dimensional complete Riemannian manifold and fix a point \(p \in M\). Suppose that the Ricci curvature tensor of \(M\) at any point \(x\) is bounded below by \((m-1)K(r(p,x))\) for some function \(K\) depending on the distance from \(p\) denoted \(r:=r(p,x)\). If \(J(\theta,r)d\theta\) is the area element of \(\partial B_p(r)\) as defined above and \(\bar{J}(r)\) is the solution of the ordinary differential equation \[\bar{J}'' = \frac{m-2}{m-1}(\bar{J}')^2\bar{J}^{-1} - (m-1)K\bar{J}\] with initial conditions \[\begin{aligned} &\bar{J}(r) \sim r^{m-1} \\ &\bar{J}'(r) \sim (m-1)r^{m-2}\end{aligned}\] as \(r\to0\), then within the cut-locus of \(p\), the function \(\frac{J(\theta,r)}{\bar{J}(r)}\) is a nonincreasing function of \(r\). Denoting \(\bar{H}(r) = \frac{\bar{J}'}{\bar{J}}\), then \(H(\theta,r)\leq\bar{H}(r)\) whenever \((\theta,r)\) is within the cut-locus of \(p\). In particular, if \(K\) is a constant, then \(\bar{J}d\theta\) corresponds to the area element of the sphere of radius \(r\) in the simply connected space form of constant curvature \(K\).

Proof. Set \(f = J^{\frac{1}{m-1}}\), then substituting into (3.1) and (3.2), we have

\[ \begin{aligned} f' = \left(J^{\frac{1}{m-1}}\right)' &= \frac{1}{m-1}\left(J^{\frac{1}{m-1}-1}\right)J' \nonumber \\ &= \frac{1}{m-1}J^{\frac{1}{m-1}-1}HJ \nonumber \\ &= \frac{1}{m-1}HJ^{\frac{1}{m-1}} = \frac{1}{m-1}Hf, \end{aligned} \](3.4)

and

\[ \begin{aligned} f'' &= \left(J^{\frac{1}{m-1}}\right)'' \nonumber \\ &= \left(\frac{1}{m-1}\left(J^{\frac{1}{m-1}-1}\right)J'\right)' \nonumber \\ &= \frac{1}{m-1}\left(\left(\frac{1}{m-1}-1\right)J^{\frac{1}{m-1}-2}(J')^2+J^{\frac{1}{m-1}-1}J''\right) \nonumber \\ &= \frac{1}{m-1}\left(\left(\frac{1}{m-1}-1\right)J^{\frac{1}{m-1}-2}H^2J^2+J^{\frac{1}{m-1}-1}J''\right) \nonumber \\ &\leq \frac{1}{m-1}\left(\left(\frac{1}{m-1}-1\right)J^{\frac{1}{m-1}}H^2+J^{\frac{1}{m-1}-1}\left(\frac{m-2}{m-1}H^2J-R_{rr}J\right)\right) \nonumber \\ &= \frac{1}{m-1}\frac{2-m}{m-1}fH^2 + \frac{1}{m-1}\frac{m-2}{m-1}fH^2 - \frac{1}{m-1}R_{rr}f \nonumber \\ &= \frac{-1}{m-1}R_{rr}f \nonumber \\ &= -Kf. \end{aligned} \](3.5)

The initial conditions correspondingly become \[\begin{aligned} &f(\theta,0) = 0 \\ &f'(\theta, 0) = 1.\end{aligned}\] Let \(\bar{f} = \bar{J}^{\frac{1}{m-1}}\) be the corresponding function defined using \(\bar{J}\). Then by the assumption that \(\bar{J}\) is the solution to the ODE, we have that \[\bar{f}'' = -K\bar{f}, \quad \bar{f}(0) = 0, \quad \bar{f}'(0) = 1.\] If \(K\) is a constant, then we have several cases:

If \(K = 0\), then \(\bar{f}'(r) = 1\) everywhere and so \(\bar{f}(r) > 0\) for all \(r \in (0,\infty)\).
If \(K \leq 0\), then \(\bar{f}'(r) \geq 1\) everywhere and so \(\bar{f}(r) > 0\) for all \(r \in (0,\infty)\).
If \(K > 0\), then \(\bar{f}'(r)\) is decreasing from an initial value of 1, so there is some \(a > 0\) such that \(0 < \bar{f}'(r) < 1\). Then \(\bar{f}(r) > 0\) for all \(r \in (0, a)\).

In any case we can suppose that there is some \(a > 0\) for which \(\bar{f}(r) > 0\) for \(r \in (0, a)\). Hence we can define \[F(\theta,r) = \frac{f(\theta,r)}{\bar{f}(r)}, \quad \text{for } r \in (0,a).\] Then by the quotient rule we have \[F' = \bar{f}^{-2}(f'\bar{f}-f\bar{f}'),\] and computing we have

\[ \begin{aligned} F'' &= \bar{f}^{-4}(\bar{f}^2(f''\bar{f}-f\bar{f}'') - 2\bar{f}\bar{f}'(\bar{f}f'-f\bar{f}')) \nonumber \\ &= \bar{f}^{-2}(f''\bar{f}-f\bar{f}'')-2\bar{f}'\bar{f}^{-1}(\bar{f}^{-2}(\bar{f}f'-f\bar{f}')) \nonumber \\ &= \bar{f}^{-2}(f''\bar{f}-f\bar{f}'') - 2\bar{f}'\bar{f}^{-1}F' \nonumber \\ &= \bar{f}^{-2}(f''\bar{f}+fK\bar{f}) - 2\bar{f}'\bar{f}^{-1}F' \nonumber \\ &= \bar{f}^{-1}(f''+K\bar{f})-2\bar{f}'\bar{f}^{-1}F' \nonumber \\ &\leq \bar{f}^{-1}(-Kf+Kf) - 2\bar{f}'\bar{f}^{-1}F' \nonumber \\ &= -2\bar{f}'\bar{f}^{-1}F', \end{aligned} \](3.6)

where we used the fact that \(\bar{f}'' = -K\bar{f}\) in the 4th equality above and the fact that \(f'' \leq -Kf\) in the inequality above. Hence we have that

\[ \begin{aligned} (\bar{f}^2F')' = (\bar{f}^2)'F' + \bar{f}^2F'' &= 2\bar{f}\bar{f}'F' + \bar{f}^2F'' \nonumber \\ &= \bar{f}^2(2\bar{f}'\bar{f}^{-1}F' + F'') \nonumber \\ &\leq \bar{f}^2(2\bar{f}'\bar{f}^{-1}F' - 2\bar{f}'\bar{f}^{-1}F') = 0. \end{aligned} \](3.7)

Let \(0 < \varepsilon < r\). Then using inequalities (3.6) and (3.7) and integrating from \(\varepsilon\) to \(r\) and using the Fundamental Theorem of Calculus we have that \[\begin{aligned} F'(r) &\leq F'(\varepsilon)\bar{f}^2(\varepsilon)\bar{f}^{-2}(r) \\ &= \bar{f}^{-2}(\varepsilon)(f'(\varepsilon)\bar{f}(\varepsilon)-f(\varepsilon)\bar{f}'(\varepsilon))\bar{f}^2(\varepsilon)\bar{f}^{-2}(r) \\ &= (f'(\varepsilon)\bar{f}(\varepsilon)-f(\varepsilon)\bar{f}'(\varepsilon))\bar{f}^{-2}(r),\end{aligned}\] and letting \(\varepsilon \to 0\) by the initial conditions we have that \(f(\varepsilon) \to 0\), \(\bar{f}(\varepsilon) \to 0\), so the right-hand side above goes to 0. That is, \[F'(r)\leq 0\] which implies that \(\bar{f}f'-\bar{f}'f \leq 0\). Then substituting \(f' = \frac{1}{m-1}Hf\), \(\bar{f}'=\frac{1}{m-1}\bar{H}\bar{f}\) we therefore can conclude that \[H(\theta,r)\leq\bar{H}(r), \quad \text{for } r \in (0,a).\] Finally, the nonincreasing property of \(F\) then also implies that \((F(r))^{m-1} = \frac{J(\theta,r)}{\bar{J}(r)}\) is nonincreasing on the same interval and we are done.

3.2 Myer’s Theorem

Theorem 2 (Myer’s Theorem). Let \(M\) be an \(m\)-dimensional complete Riemannian manifold with Ricci curvature bounded from below by \[R_{ij} \geq (m-1)K\] for some positive constant \(K > 0\). Then \(M\) must be compact with diameter \(d\) bounded from above by \[d\leq \frac{\pi}{\sqrt{K}}.\]

Proof. Under the same assumptions as in Bishop’s Comparison Theorem, if \(K\) is a constant, then we can compute the area element and mean curvature of a space form with constant sectional curvature \(K\) explicitly: \[\bar{J}(r) = \begin{cases} \left(\frac{1}{\sqrt{K}}\right)^{m-1}\sin^{m-1}\left(\sqrt{K}r\right) &\text{for } K > 0, \\ r^{m-1} &\text{for } K = 0, \\ \left(\frac{1}{\sqrt{-K}}\right)^{m-1}\sinh^{m-1}\left(\sqrt{-K}r\right) &\text{for } K < 0, \end{cases}\] and by the conclusion of Bishop’s Comparison Theorem we get that \(\frac{J(\theta,r)}{\bar{J}(r)}\) is nonincreasing in \(r\) and we can estimate: \[H(r) \leq \frac{\bar{J}'(r)}{\bar{J}(r)} = \begin{cases} \left(\frac{1}{\sqrt{K}}\right)^{m-1}\cot^{m-1}\left(\sqrt{K}r\right) &\text{for } K > 0, \\ (m-1)r^{-1} &\text{for } K = 0, \\ \left(\frac{1}{\sqrt{-K}}\right)^{m-1}\coth^{m-1}\left(\sqrt{-K}r\right) &\text{for } K < 0. \end{cases}\] Then under our assumptions we are interested in the \(K > 0\) case, and the estimate on \(H\) implies that there must be a cut-point (that is, a point \(q\) which makes \(M\setminus\{q\}\) disconnected) along any geodesic which has length \(\frac{\pi}{\sqrt{K}}\) since at that distance \(\bar{J}\left(\frac{\pi}{\sqrt{K}}\right)\) vanishes. Hence, \(M\) must have diameter \(d\) bounded from above by \(\frac{\pi}{\sqrt{K}}\). Then from a basepoint \(p \in M\), by the corollary of the Hopf-Rinow Theorem given in Section 1 above, any other point \(q\) can be connected to \(p\) can be connected to \(p\) by a geodesic segment of length at most \(\frac{\pi}{\sqrt{K}}\). This implies that the exponential map at \(p\), \(\exp_p:\bar{B}_{\frac{\pi}{\sqrt{K}}}(0) \to M\) is surjective. So \(M\) is the continuous image of a compact set and hence itself is compact.

Also note that this result implies that the universal cover of \(M\) is also compact and hence \(M\) has finite fundamental group.

3.3 Other Further Results

While Myer’s theorem can be proven using more directly (again, see p.200-201 of (Lee 1997)), one advantage of developing the framework of volume comparison is that that they can be extended to prove other local-to-global results along the same vein of Myer’s theorem. In particular, these tools can be used to prove Cheng’s diameter rigity theorem ((Li 2012), p. 18), and develop further results such as the Laplacian comparison theorem, Cheeger-Gromoll’s splitting theorem and Cheng’s Eigenvalue comparison theorem ((Li 2012), Chapter 4) for Riemannian manifolds with Ricci curvature bounded from below. See (Li 2012) for further exposition of these results, which, due to length considerations, unfortunately could not be included here.