global-analysis-and-the-banach-manifold-of-class-h1-curvers

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

preamble.tex (1946B)

 1 \documentclass[a4paper]{article}
 2 \usepackage[total={6in, 9in}]{geometry}
 3 \usepackage{amsmath, amssymb, amsthm, stmaryrd, mathrsfs, gensymb, dsfont}
 4 \usepackage{mathtools, adjustbox}
 5 \usepackage[scr=esstix,cal=boondox]{mathalfa}
 6 \usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize}
 7 \usepackage{hyperref}
 8 \usepackage[backend=biber]{biblatex}
 9 \usepackage{pgfplots, tikz, tikz-cd}
10 \usepackage{graphicx, wrapfig}
11 \usepackage[normalem]{ulem}
12 
13 % Configure graphics
14 \pgfplotsset{compat=1.18}
15 
16 % Use \blacksquare for \qed
17 \renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
18 
19 % Get propper inequality symbols
20 \renewcommand{\leq}{\leqslant}
21 \renewcommand{\le}{\leqslant}
22 \renewcommand{\geq}{\geqslant}
23 \renewcommand{\ge}{\geqslant}
24 \renewcommand{\preceq}{\preccurlyeq}
25 \renewcommand{\succeq}{\succcurlyeq}
26 
27 % Add missing arrows
28 \newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow}
29 
30 % Fix the subset symbol
31 \renewcommand{\subset}{\subseteq}
32 
33 % Display long arrows instead of short ones
34 \renewcommand{\to}{\longrightarrow}
35 \renewcommand{\mapsto}{\longmapsto}
36 
37 % Absolute value and vector norm
38 \newcommand{\abs}[1]{\left|\nobreak#1\nobreak\right|}
39 \newcommand{\norm}[1]{\left\lVert\nobreak#1\nobreak\right\rVert}
40 
41 \newcommand{\dt}{\mathrm{d} t}
42 \newcommand{\dd}{\mathrm{d}}
43 
44 % Isomorphism arrow
45 \newcommand{\isoto}{\xlongrightarrow{\sim}}
46 
47 % Configure how link look
48 \hypersetup{
49   colorlinks,
50   citecolor=black,
51   filecolor=black,
52   linkcolor=black
53 }
54 
55 % Configure the enumerate environment to use bold roman numerals
56 \setenumerate[0]{label={\normalfont\bfseries(\roman*)}}
57 
58 % Useful theorem definitions
59 \newtheorem{theorem}{Theorem}[section]
60 \newtheorem{lemma}{Lemma}[section]
61 \newtheorem{corollary}{Corollary}[section]
62 \newtheorem{proposition}{Proposition}[section]
63 \theoremstyle{remark}
64 \newtheorem*{note}{Remark}
65 \theoremstyle{definition}
66 \newtheorem{definition}{Definition}[section]
67 \newtheorem{example}{Example}[section]
68