Riemannian Geometry course project on the manifold H¹(I, M) of class H¹ curves on a Riemannian manifold M and its applications to the geodesics problem
Commit
1dfc5bb9948bd9e62f74b8305b0756a4a442c006 Parent
1126d5038b8c8e55cf01af2d68aa522c0a4cb81c Author
Pablo <pablo-escobar@riseup.net >
Date
Sat, 27 Aug 2022 19:26:00 +0000
Fixed a bunch of typos
Replaced "this" with "these" in the appropriate places
Diffstat
3 files changed, 22 insertions, 21 deletions
diff --git a/sections/applications.tex b/sections/applications.tex
@@ -151,7 +151,7 @@ We can now show\dots
\end{theorem}
\begin{proof}
- To see that this are submanifolds, it suffices to note that \(\Omega_{p q}
+ To see that these are submanifolds, it suffices to note that \(\Omega_{p q}
M\) and \(\Lambda M\) are the inverse images of the closed submanifolds
\(\{(p, q)\}, \{(p, p) : p \in M\} \subset M \times M\) under the submersion
\((\sigma, \tau) : H^1(I, M) \to M \times M\).
@@ -424,7 +424,7 @@ Once again, the first part of this proposition is a particular case of a
broader result regarding the space of curves joining submanifolds of \(M\): if
\(N \subset M\) is a totally geodesic manifold of codimension \(1\) and
\(\gamma \in H_{N, \{q\}}^1(I, M)\) is a critical point of the restriction of
-\(E\) then \(A_\gamma = \operatorname{Id} + K_\gamma\). This results aren't
+\(E\) then \(A_\gamma = \operatorname{Id} + K_\gamma\). These results aren't
that appealing on their own, but they allow us to establish the following
result, which is essential for stating Morse's index theorem.
@@ -557,7 +557,7 @@ of the classical formulation: we can find curves \(\eta = i(v)\) shorter than
\(\gamma\) already in an \(H^1\)-neighborhood of \(\gamma\).
This concludes our discussion of the applications of our theory to the
-geodesics problem. We hope that this short notes could provide the reader with
+geodesics problem. We hope that these short notes could provide the reader with
a glimpse of the rich theory of the calculus of variations and global analysis
at large. We once again refer the reader to \cite[ch.~2]{klingenberg},
\cite[ch.~11]{palais} and \cite[sec.~6]{eells} for further insight on modern
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -38,9 +38,9 @@ section functor, such as smooth sections or Sobolev sections -- notice that by
taking \(E = M \times N\) the manifold \(\Gamma(E)\) is naturally identified
with a space of functions \(M \to N\), which gets us back to the original case.
-In this notes we hope to provide a very brief introduction to modern theory the
-calculus of variations by exploring one of the simplest concrete examples of
-the previously described program: we study the differential structure of the
+In these notes we hope to provide a very brief introduction to modern theory
+the calculus of variations by exploring one of the simplest concrete examples
+of the previously described program: we study the differential structure of the
Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a finite-dimensional
Riemannian manifold \(M\), which encodes the solution to the \emph{classic}
variational problem: that of geodesics. Hence the particular action functional
@@ -69,7 +69,7 @@ We should point out that we will primarily focus on
the broad strokes of the theory ahead and that we will leave many results
unproved. The reasoning behind this is twofold. First, we don't want to bore
the reader with the numerous technical details of some of the constructions
-we'll discuss in the following. Secondly, and this is more important, this
+we'll discuss in the following. Secondly, and this is more important, these
notes are meant to be concise. Hence we do not have the necessary space to
discuss neither technicalities nor more involved applications of the theory we
will develop.
@@ -77,7 +77,7 @@ will develop.
In particular, we leave the intricacies of Palais' discussion of condition (C)
-- which can be seen as a substitute for the failure of a proper Hilbert space
to be locally compact \cite[ch.~2]{klingenberg} -- and its applications to the
-study of closed geodesics out of this notes. As previously stated, many results
+study of closed geodesics out of these notes. As previously stated, many results
are left unproved, but we will include references to other materials containing
proofs. We'll assume basic knowledge of differential and Riemannian geometry,
as well as some familiarity with the classical theory of the calculus of
@@ -101,7 +101,7 @@ turns out that calculus has nothing to do with \(\mathbb{R}^n\).
What we mean by this last statement is that none of the fundamental ingredients
of calculus -- the ones necessary to define differentiable functions in
\(\mathbb{R}^n\), namely the fact that \(\mathbb{R}^n\) is a complete normed
-space -- are specific to \(\mathbb{R}^n\). In fact, this ingredients are
+space -- are specific to \(\mathbb{R}^n\). In fact, these ingredients are
precisely the features of a Banach space. Thus we may naturally generalize
calculus to arbitrary Banach spaces, and consequently generalize smooth
manifolds to spaces modeled after Banach spaces. We begin by the former.
@@ -155,7 +155,7 @@ chain rule}.
\]
\end{lemma}
-As promised, this simple definitions allows us to expand the usual tools of
+As promised, these simple definitions allows us to expand the usual tools of
differential geometry to the infinite-dimensional setting. In fact, in most
cases it suffices to simply copy the definition of the finite-dimensional case.
For instance, as in the finite-dimensional case we may call a map between
@@ -207,7 +207,7 @@ in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
vector space, and this topology is independent of this choice\footnote{In
general $T_p M$ is not a normed space, since the norms induced by two
distinct choices of chard need not to coincide. Nevertheless, the topology
- induced by this norms is the same.}.
+ induced by these norms is the same.}.
\end{proposition}
\begin{proof}
@@ -276,9 +276,9 @@ explicitly stated otherwise. Speaking of examples\dots
operators \(H \to H\) \cite[p.~4]{unitary-group-strong-topology}.
\end{example}
-This last two examples are examples of Banach Lie groups -- i.e. Banach
+These last two examples are examples of Banach Lie groups -- i.e. Banach
manifolds endowed with a group structure whose group operations are smooth.
-Perhaps more interesting to us is the fact that this are both examples of
+Perhaps more interesting to us is the fact that these are both examples of
function spaces. Having reviewed the basics of the theory of Banach manifolds
we can proceed to our in-depth exploration of a particular example, that of the
space \(H^1(I, M)\).
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -301,7 +301,7 @@ of this proof is showing that the transition maps
\to H^1(\eta^* TM)
\]
are diffeomorphisms, as well as showing that \(H^1(I, M)\) is separable. We
-leave this details as an exercise to the reader -- see theorem 2.3.12 of
+leave these details as an exercise to the reader -- see theorem 2.3.12 of
\cite{klingenberg} for a full proof.
It's interesting to note that this construction is functorial. More
@@ -345,7 +345,7 @@ The space \(H^1(E)\) is modeled after the Hilbert spaces \(H^1(F)\) of class
of a vector bundle -- the so called \emph{vector bundle neighborhoods of
\(E\)}. This construction is highlighted in great detail and generality in the
first section of \cite[ch.~11]{palais}, but unfortunately we cannot afford such
-a diversion in this short notes. Having said that, we are now finally ready to
+a diversion in these short notes. Having said that, we are now finally ready to
layout the Riemannian structure of \(H^1(I, M)\).
We are finally ready to discuss some applications.
@@ -357,7 +357,7 @@ at its tangent bundle. Notice that for each \(\gamma \in {C'}^\infty(I, M)\)
the chart \(\exp_\gamma^{-1} : U_\gamma \to H^1(\gamma^* TM)\) induces a
canonical isomorphism \(\phi_\gamma = \phi_{\gamma, \gamma} : T_\gamma H^1(I,
M) \isoto H^1(\gamma^* TM)\), as described in
-proposition~\ref{thm:tanget-space-topology}. In fact, this isomorphisms may be
+proposition~\ref{thm:tanget-space-topology}. In fact, these isomorphisms may be
extended to a canonical isomorphism of vector bundles, as seen in\dots
\begin{lemma}\label{thm:alpha-fiber-bundles-definition}
@@ -421,10 +421,11 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
which takes \(\varphi^{-1}(X, Y) \in T_{exp_\gamma(X)} H^1(I, M)\) to
\(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
enought patience, one can deduce from the fact that \(\varphi_\gamma\) and
- \(\psi_{1, \gamma}^{-1}\) are charts that this maps agree in the intersection
- of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma
- H^1(I, M))\), so that they may be glued together into a global smooth map
- \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^* TM)\).
+ \(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
+ intersections of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times
+ T_\gamma H^1(I, M))\), so that they may be glued together into a global
+ smooth map \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^*
+ TM)\).
Since this map is a fiber-preserving, fiber-wise linear local diffeomorphism,
this is an isomorphism of vector bundles.
@@ -532,7 +533,7 @@ words, we'll show\dots
simply by \(\frac\nabla\dt X\).
\end{proposition}
-The proofs of this last two propositions were deemed too technical to be
+The proofs of these last two propositions were deemed too technical to be
included in here, but see proposition 2.3.16 and 2.3.18 of \cite{klingenberg}.
We may now finally describe the canonical Riemannian metric of \(H^1(I, M)\).